Hua C., Wong R. Differential Equations and Asymptotic Theory in Mathematical Physics
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10.
P.
Deift and
X.
Zhou,
A
steepest descent method
for
oscillatory Riemann-
Hilbert problems, Applications
for
the MKdV equation,
Ann. Math.
137
(1993), 295-368.
11.
R. B. Dingle and G.
J.
Morgan,
WKB
methods
for
dzfference equtions
I,
Appl.
Sci. Res.
18
(1967), 221-237.
12.
T.
Kriecherbauer and K. T-R McLaughlin,
Strong asymptotics
of
polynomials
orthogonal with respect to Freud weights,
Internat. Math. Res. Notes
(1999),
13.
A. B.
J.
Kuijlaars and K. T-R McLaughlin,
Riemann-Hilbert analysis
for
Laguerre polynomials with large negative parameter,
Comput. Math. Funct.
Theory
1
(2001), 205-233.
14.
A.
B.
J.
Kuijlaars and M. Vanlessen,
Universality
for
eigenvalue correla-
tions from the modified Jacobi unitary ensemble,
Internat. Math. Res. Notices
(2002), 1575- 1600.
15.
F.
W.
J.
Olver,
Asymptotics and Special Functions,
Academic Press, New
York,
1974.
(Reprinted by A. K. Peters Ltd., Wellesley,
1997).
16.
G. Szego,
Orthogonal Polynomials,
Fourth edition, Colloquium Publications,
Vol
23,
Amer. Math.
SOC.,
Providence
R. I.,
1975.
17.
W.
Van Assche,
J.
S.
Geromino, and A.
B.
J.
Kuijlaars,
Riemann-Hilbert
problems
for
multiple orthogonal polynomials,
pp.23-50
in “NATO AS1 Spe-
cial Functions
2000”
(J.
Bustoz et. al., eds.), Kluwer Academic Publishers,
Dordrecht
2001.
18.
Z.
Wang,
Asymptotic expansions
for
second order linear difference equations
with a
turning
point,
Ph.D. Thesis, City University
of
Hong Kong,
2001.
19.
Z.
Wang and R. Wong,
Uniform asymptotic expansion
of
J,(va)
via a dif-
ference equation,
Numer. Math.
91
(2002), 147-193.
20.
Z. Wang and R. Wong,
Asymptotic expansions
for
second-order linear differ-
ence equations with a turning point,
Numer. Math.
94
(2003), 147-194.
21.
Z.
Wang and R. Wong,
Linear difference equations with transition points,
Math. Comp., to appear.
22.
R. Wong,
Asymptotic Approximations
of
Integrals,
Academic Press, Boston,
1989.
(Reprinted by SIAM, Philadelphia, PA,
2001.)
23.
R. Wong and H. Li,
Asymptotic expansions
for
second-order linear difference
equations,
J.
Comput. Appl. Math.
41
(1992), 65-94.
24.
R. Wong and
H.
Li,
Asymptotic expansions
for
second-order linear difference
equations
11,
Stud. Appl. Math.
87
(1992), 289-324.
299-333.
A
PERTURBATION MODEL
FOR
THE GROWTH
OF
TYPE
111-V
COMPOUND CRYSTALS
c.
SEAN
BOHUN:
IAN
FRIGAARD:
HUAXIONG
HUANG)
&
SHUQING
LIANG~
In this paper we describe the basic idea
of
a semi-analytical approach
for
computing
the temperature and thermal stress inside
a
type
111-V
compound grown with the
Czochralski technique. An analysis
of
the growing conditions indicates that the
crystal growth occurs on the conductive time scale. A perturbation method
for
the temperature field is developed using the Biot number
as
a
(small) expansion
parameter whose zeroth order solution is one-dimensional (in the axial direction)
and is obtained for
a
cylindrical and a conical crystal. Under typical growth
conditions,
a
parabolic temperature profile in the radial direction is shown to arise
naturally
as
the first order correction. As
a
result, the thermal stress
is
obtained
explicitly and its magnitude is shown to depend on the zeroth order temperature
and Biot number.
1.
Introduction
The Czochralski method is the most popular technique for growing the
single large crystals used by the semi-conductor and related industries. By
dipping a small seed crystal into a pool of molten material in the crucible
and carefully controlling the heat balance inside the grower,
a
large crystal
can be grown by pulling the crystal away from the melt in a slow and
steady fashion.
The pulling rod and the crucible are normally rotated
in the opposite directions during the growth period. Delicate control is
often needed to maintain the crystal quality and a slight change of growth
condition may result in defect formation inside the crystal. With care,
a
*department
of
mathematics, Pennsylvania state university, mont alto, pa 17237.
csblS(0psu.edu
tdepartment of mathematics, university
of
british Columbia, Vancouver, bc v6t 122.
f
rigaardhath. ubc
.
ca
tdepartment
of
mathematics and statistics, york university, toronto, Ontario m3j lp3.
hhuang(0yorku.ca
§department
of
mathematics and statistics, york university, toronto, Ontario m3j lp3.
sqliangkathstat. yorku. ca
263
264
pure (‘defect free’) crystal can be obtained routinely when the size of the
crystal does not exceed a critical size.
Due to the seemingly complex nature of the thermal, structural and dy-
namical coupling
of
the molten material, the crystal, the crucible, the gas
chamber and other parts of the growth, considerable efforts have been de-
voted to the laboratory experiments and to modelling and simulations of the
growth environment over the past several decades. As a result, there exists
an extensive literature, mostly in engineering fields. These studies cover
a
wide spectrum of areas, from decoupled one or two dimensional simulations
to fully coupled three-dimensional computations
3,4,9110,13114915.
Most of the
studies rely heavily on computer simulation since the fully coupled system
can not be solved otherwise. These investigations have generated useful in-
formation including temperature distribution, crystal-melt interface shape,
and melt flow patterns inside the crucible. By comparison, until recently
much less attention has been paid to the modelling of defects inside the
crystal and main factors which determine the formation of defects
16.
In this paper, we present a perturbation approach to study the temper-
ature field inside the crystal and related thermal stress. It is believed that
the defects formation can be related to the excessive thermal stress above
some critical value (see
11>18&21>731591g
and references therein). Therefore,
some analysis on the growth factors which determine the stress level will be
extremely useful for crystal growers using different operating conditions for
less well-known type 111-V crystals such as the Indium Antimonide (InSb)
compound. While the basic solution remains the same for these compounds,
we will focus on InSb crystals in the remainer of the paper.
By examining the physical process and parameter values of the growth
environment closely, we are able to identify the main features associated
with InSb crystals. In particular, we found that the temperature field is
dominated by the lateral flux through the crystal-gas surface, characterized
by the non-dimensional Biot number. The value of the Biot number is small
under the growth condition for InSb crystals, which suggests an asymptotic
expansion of the solution with respect to the Biot number.
As
a result, an-
alytical solutions could be obtained for the pseudo-steady case, which is
another main feature of the temperature field inside the crystal. This is
similar to the growth of other crystals where the pseudo-steady assumption
has been discussed in detail
6.
Even for the fully unsteady case, the asymp-
totic expansion results in a system of one-dimensional equations and the
thermal stress can be obtained explicitly in an analytical form, under the
plane strain assumption. To simplify the presentation, we have used a sim-
265
plified model for the melt and gas flows. However, the asymptotic solution
developed here is still valid and can be incorporated with more realistic
models for the melt and gas flows. Compared to most of the previous work,
the results of this study are different since an explicit form for the stress
is obtained. Formulated in a non-dimensional form, the dependence of the
stress level on the Biot number is useful for crystal growers when larger
crystals are grown. Since the Biot number is proportional to the product
of the heat transfer coefficient and the mean crystal radius, it is obvious
that one should try to reduce the heat flux via the lateral surface when a
crystal of larger radius is grown.
The rest of the paper is organized as follows. In Section
2,
we will
present the mathematical model and dimensional analysis. Asymptotic
solutions are given in Section
3.
Thermal stress is discussed in Section
4.
We will conclude the paper with a brief summary and discussion on future
directions in Section
5.
2.
Mathematical Model and Dimensional Analysis
The overall aim of the paper is to derive a realistic but simplified model of
InSb crystal growth. Figure
1
illustrates the profile
of
a
typical crystal and
the coordinate system, which is fixed to the top of the growing crystal.
Within the crystal
R,
the temperature
T(x,
t)
satisfies the heat equation
XER,
t>O
where
p,,
c,
and
k,
are the density, specific heat and thermal conductivity
of the crystal, (i.e. the solid phase), respectively. The lateral surface of the
crystal is denoted
rg
and is subjected to cooling from the circulating cham-
ber gases and radiative heat loss. Although radiation
is
not insignificant,
for simplicity we model both effects through
a
simple Newtonian cooling
law:
aT
an
-ks-
=
hgs(T
-
Tg),
x
E
rg.
Here we assume that the heat transfer coefficient,
hgs,
incorporates both
convective and radiative heat transfer, (the latter via linearisation). The
top
of
the crystal is fixed at
t
=
0
where we also invoke a Newtonian cooling
law
266
j
k
n
C
=
2zR
A
=
nR2
Figure
1.
Shown is a typical crystal at some time
t
during
a growth
run
with a newly
solidified portion at
z
=
S(t).
The coordinate system is chosen
so
that the top of the
crystal remains at
z
=
0
and the solidijcation front grows downwards
in
the positive
z
direction. The radial profile is given by
R(z)
and the crystal length is
S(t).
Finally, the
heat transfer coefficient hgs(z) may be a function of the axial position
z.
in the case that the radius at
z
=
0
is assumed to be non-zero. Here
hch
represents the heat transfer coefficient for the seed-chuck connection and
Tch
is the chuck temperature.
The crystal-melt interface is denoted
rs
and is where
T
=
T,,
the
melting temperature. The interface of the phase transition is thus implic-
itly defined from the temperature field. Explicity we denote the melting
isotherm by
z
-
S(x,t)
=
0,
x
E
rs.
(4)
The motion of the interface of the phase transition is governed by the Stefan
condition
where
L
is the latent heat and
91
is the heat flux from the melt. The speed
aS/at
above is the speed at which
5’
moves in the direction of the outward
normal
n.
267
Data
Symbol Value
Mean crystal radius
R
0.03
m
Final crystal length
2
0.30
m
Characteristic growth rate
V
3
x
lop6
m/s
Ambient gas temperature
Tg
600
K
Solid Properties at
T
=
T,
Melting temperature
T,
798.4
K
Density
p,
5.64
x
lo3
kg/m3
Thermal conductivity
k,
4.57
W/m
K
Heat capacity
psc,
1.5
x
lo6
J/m3
K
Latent heat of fusion
L
2.3
x
lo5
J/kg
Heat Transfer Coefficients
Crystal-Gas
h,,
1-4W/m2K
Growing Properties
-
2.1.
Non-
dimensionalitation
Typical values are shown in Table 1. Using these values we can deduce that
the main thermal gradients are in the radial direction, due to, gas cooling
at the surface, and that these drive the vertical thermal gradients. The
conductive timescale is generally much shorter than the growth timescale,
and this indicates that the process is pseudo-steady.
These observations motivate our scaling below. For simplicity, we start
by assuming an axisymmetric model, although the crystal cross-section is
not in fact circular. The other assumptions that we make here, for simplicity
only, are that the heat transfer coefficient
h,,
and the gas temperature
Tg
are constant. In reality there will be local variations along the crystal
surface, but
in
any case these require a more detailed analysis of the gas
flows in order to be properly evaluated.
We define the Biot number by
and using the parameter values in Table
2,
we find
E
N
0.026
<<
1. We seek
an asymptotic expansion in terms of
E.
With this in mind we adopt the
268
following scalings:
StR2p,c,
A
L
t= t,
St
=
-
k?
c,AT
Here variables with hats
(
)
are the non-dimensional ones. In terms of
these variables the heat equation in the crystal
(1)
becomes
E
1
with boundary conditions
(2)-(4)
-0t
St
=
-(re,),
r
+
&,,,
XER,
t>O
(7)
-0,
+
EO,R’(Z)
=
E
[l
+
E(R’(%))2]
1/2
0,
x
E
rg,
z
=
0,
0
=
1,
x
E
rs,
(8)
0,
=
S(@
-
@&),
where
S
=
c1/2hch/hgs.
The hats have been dropped for brevity. The
crystal/melt interface advances according to the Stefan condition
(5)
which
in non-dimensional coordinates becomes
1
0,
-
-S,O,
=
(1
+
$)
1/2
(y
+
St),
E
where
(9)
which is the non-dimensional heat flux in the liquid across the crystal/melt
interface. Typically the value of
y
is determined by the characteristics of
the melt, which is controlled by the growth conditions such as the heater
temperature, crucible shape, etc.
To
simplify the discussion here, we will
assume that the value of
y
can be chosen. In
2,
a simple model is used
to determine
y
and work
is
currently underway to develop a fully-coupled
model.
Note that we have chosen the rate of solidification to define the charac-
teristic time scale. The Stefan number, St, gives the ratio of this character-
istic solidification time scale to the time scale associated with conductive
heat loss through the crystal side surface. Based on the parameter values
in Table
2,
we have St
x
4.3,
which suggests that the conductive scale is
small and the temperature inside the crystal is steady on the growth time
scale.
269
3.
Perturbation Solution
We now seek to approximate the scaled model in Section
2.1
via a straight-
forward perturbation expansion. In turn, this perturbation model will form
the basis for
a
numerical solution. Since St is neither small nor too large
under the current growth conditions it is retained as a parameter. Equa-
tions
(7)
&
(8)
strongly suggest that the temperature
0
is independent of
r
to leading order. If true then the crystal/melt interface
S
is also indepen-
dent of
T
to leading order, and we see that this is consistent in
(9)
with the
growth being driven primarily by the vertical gradients. These observations
motivate the following approximations:
(11)
0-0o(z,t)+~O1(r,z,t) +~~02(r,z,t)
+...
S
-
So@)
+
E&(r,t)
+
e2S2(r,t)
+
. . . .
We substitute them into the scaled model, expand in powers of
E,
simplify
and collect terms. The resulting field equations to first order are:
(12)
XER,
t>0,
XER,
t>0,
(13)
-
R’Oo,=
=
-00,
r
=
R(z),
(14)
r
=
R(z).
U5)
1
ld
r dr
1
18
-%,t
=
--(~@2,~)
+
01,~~~
St
r dr
-
St
@o,t
=
--(r@l,T)
+
00,221
where the boundary condition on the lateral surface becomes
1
2
02,T
-
R’01,,
+
-Rt200
=
-01,
Continuing this procedure for the remaining conditions,
at
the top of the
crystal one has
00,z
=
S(00
-
Och),
@I,*
=
601,
z
=
0,
z
=
0,
and at the solid-liquid interface
00
=
1,
sloo,2
+
01
=
0,
z
=
So(t),
z
=
So(t).
Finally, the evolution of the interface is governed by
Y
+
So,t
=
~o,Z12=~o(t)
,
so(o)
=
Zseed7
(I6)
S,(r,O)
=
0.
(17)
We note that
Zseed
is the non-dimensional length of the seed. In addition
there will be symmetry conditions at
r
=
0
for
0k,
Sk,
k
=
0,l.
270
3.1.
Resolution
of
the zeroth order model
Integrating
(12)
once and imposing the symmetry condition
01,~
=
0
at
r
=
0,
we have:
and applying
(14)
at
r
=
R
gives the zeroth order problem:
1
2
--Oo,t
=
@O,~Z
+
-
(R’@o,Z
-
00)
,
St
R
0
<
z
<
So(t),
t
>
0,
(18)
@o,z
=
S(@O
-
Qch),
Z
=
0,
(19)
00
=
1,
Z
=
So(t)
(20)
where the advance of
So@)
is coupled to the thermal gradients via (16).
Equation
(18)
is parabolic and involves only the heat fluxes along the
length of the crystal. With the chosen expansion we see that at zeroth order
the temperature field has no radial dependence. In addition, we can see that
the thermal gradients,
as
discussed previously, are caused by cooling effects
at
the surface. Also notice that expression (16) illustrates that the chosen
time scale balances the growth. The resultant appearance
of
l/St
<
1
in
(18)
suggests that thermal transients in the bulk
of
the crystal are not
as
important
as
the growth transient. The limit as St
+
00
leads naturally
to
a’
pseudo-steady leading order model, in which time dependency only
enters the thermal model through the growth, i.e. we also solve:
2
00,Zz
+
-
R
(R’@o,Z
-
00)
=
0,
0
<
2
<
So(t)
(21)
with the boundary conditions (19)
&
(20) with the growth of
So(t)
given
by (16) as the pseudo-steady limit.
We note that properly it is necessary to close the model by relating
growth in
S
to that in
R.
To do this we must model the crystal withdrawal
from the crucible, formation
of
the meniscus in the holm region and coupling
of
S
and
R.
It has been shown in
l7
that the growth angle
is
related to the
capillary height for large Bond number growth. In principle, crystals with
desirable shapes can be grown by adjusting the pulling rate. Therefore, to
simplify the computation, we impose
a
geometry
R(z)
on the model. This
approach has the advantage
of
allowing us to investigate the thermal fields
and associated stresses that develop for
a
particular observed shape. We
start by exploring two special cases for which an analytic solution may be
computed to the pseudo-steady model.
271
3.1.1.
Constant radius crystals
In this case we take
R(z)
=
1
and
(21)
becomes simply
0
=
Oo,zz
-
200,
0
<
z
<
So(t)
with boundary conditions (19)
&
(20). Solving for
00
gives:
Jz
cosh
&z
+
6
sinh
fiz
+
60,h
sinh
A(
So
-
z)
0o(z)
=
.
(22)
cosh
&'So
+
6
sinh
&SO
The crystal grows at a rate governed by the Stefan condition
(16):
fi
sinh
+
6
cosh
-
60,h
y
+
S0,t
=
Jz
cosh
&SO
+
6
sinh
&SO
The insulated chuck
(6
=
0)
and the cold chuck
(Och
=
0)
cases are easily
extracted
.
3.1.2.
Conical crystals
One source of ambiguity in the constant radius model above is the need to
specify the chuck temperature and heat transfer coefficient. In the case of a
conical'crystal, (which is closer to reality), this ambiguity is less prominent.
We assume
R(z)
=
Rseed+az
where arctana
N
O(1)
is one-half the opening
angle of the crystal when using non-dimensional units. In this case we solve
2
77
}
(23)
0
=
@O,qq
+
-(@O,q
-
OO),
00,~
=
ah(%
-
Och),
00
=
1,
Rseed
<
a277
<
Rseed
-k
a&,
t
>
0,
(r2q
=
Rseed,
a27
=
Rseed
+
aSo,
where
a2v(z)
=
Rseed
+
az.
The resulting solution takes the form of linear
combinations of modified Bessel functions
where
and