Directional melting of binary systems, as encountered during seeding in melt growth, is analyzed for concurrent compositional changes at the crystal-melt interface. It is shown that steady state conditions cannot normally be reached during seeding and that the growth interface temperature at the initial stages of seeded growth is a function of backmelt conditions. The theoretical treatment is numerically applied to Hg1-xCdXTe and Ga-doped Ge

Bourret, E. D.

Favier, J. J.

Witt, A. F.

English

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^. {NA A-CR-169072)	 SEGREGATICN DURING	 S82 -27157
' DIRECTIONAL MELTING AND ITS INFLICATICUS 09
SEEDED CRYSTAL GROWTH:
	 A 'THEORETICAL
ANALYSIS	 (;Massachusetts lust. of Tech.)
	 Unclas
1 p HC A02/MF A01
	 CSCL 20L G3/76	 23962
r SEGREGATION DURING DIRECTIONAL MELTING
AND ITS IMPLICATIONS ON SEEDED CRYSTAL GROWTH:
A THEORETICAL ANALYSIS
' E. D. Bourret, J. J. Favier" and A. F. Witt
Department of !Materials Science $ Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
ABSTRACT
Directional melting of binary systems, as encountered during
seeding in melt growth, is analyzed for concurrent compositional
changes at the crystal-melt interface.	 It is shown that steady
state conditions cannot normally be reached during seeding and
that the growth interface temperature at the initial stages of
seeded growth is a function of backmelt conditions. 	 The theoretical
treatment is numerically applied to Hg l- and Ga-doped Ge.XCdXTe
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nLaboratoire L Ctudes de Solidification (DMG), C,E.N.G.,'"Gfenb,	'', France
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1. INTRODUCTION
Directional melting, contrary to solidification, has been the subject of
only very few studies. Woodruff and Forty (1) investigated melting and
freezing of a Na-K alloy by in situ transmission UV microscopy; optical
phenomena observed in their melting experiments have been interpreted as
0
indicative for Na diffusion from the solid into the adjacent liquid phase. The
directional melting studies by Verhoeven et al. (2) in Sn-Bi, carried out at a
rate of 0.2 Vm/sec, reportedly revealed a composition gradient region of 650 Um
in the solid ahead of the receding interface. The establishment of such an
extended gradient region (boundary layer) was found by them to conflict with
theoretical prediction (1,3 um) based on the application of their steady state
segregation equation (3) for melting. It should be noted that the extended
solute depleted region in their experiments was a two-phase region with liquid
droplets appearing dispersed in the solid phase. They also report that no
solute depletion region in the solid could be observed at higher melting rates
(20 um/sec). Verhoeven and Heimes (4) provided analytical solutions for
initial melting transients with preceding solidification which consider
diffusion in both the solid and liquid phases.
All seeded crystal growth from the melt involves as a first step a varying
degree of backmelting of a crystal. This approach, generally successful in
systems with small liquidus-solidus separation such as doped semiconductors,
must, however, be expected to be problematic if binary or multicomponent systems
with large liquidus-solidus separation (Hg, l_XCdxTe, Pbl-xSnX"e, Ge-Si and the
like) are involved. Any changes in composition that occur during melting at
the crystal-melt interface are of importance since they control the growth
interface temperature and its stability. Moreover, both parameters influence
the system's behavior during equilibration (melting arrest) and may affect the
ki
Y.3.
degree of crystalline and compositional perfection achievable during ensuing
solidification. Seeded crystal growth, for exmple by the Bridgman and
Czochralski techniques, involves directional melting under boundary and
initial conditions which were not considered in the earlier treatments (q).
It is the purpose of this work to analyze the evolution of the solute
concentrations, CS(C and, C*(t), at a crystal-melt interface and the related
interface temperature: during melting, with emphasis on the initial transient
behavior. The analysis applies to binary systems in a configuration where a
crystal of composition CS in contact with a melt of composition C L is direc-
tionally melted. The theoretical treatment is carried out in non-directional
form; it is numerically applied to Hg l-xCdxTe and Ga-doped Ge to provide for
comparative analysis of systems with small and large liquidus-solidus separa-
tion.
2. Theoretical Analysis of Solute Redistribution During Directional Meltin g
2.1 Definition of boundary conditions and mass transport relationships
The analysis for directional.melting of binary solids is made with the
following assumptions:
(a) the crystal-melt interface is planar; ono-dimensional
analysis;
(b) the interface displacement R is constant;
(e) the mass transports in the liquid and in the solid
obey Pick's law;
(d) convection is neglected;
(e) the interface distribution coefficient k, taken to
be identical with the equilibrium distribution
coefficient, is a constant;
(f) the diffusion coefficients in the liquid (D
L} and
in the solid (DS) are constanti
-4-
(g) the initial length of the solid is 'larger than the
characteristic length of diffusion;
(h) the initial compositions of the solid and contacting
melt are in thermodynamic equilibrium (CS R kCh).
`	 It is recognized that a constant D is not realized during melting since
steep temperature and concentration gradients encountered across the crystal-
melt interface must affect its magnitude. A detailed determination of the
changing driving forces resulting from the existing gradients is complex
because of their interdependence. Presently, the contributions of these
driving forces are accounted for as an additional flux term, Jp(z,t), which
modifies Fick 1 s second law as follows:
V  • C(z,t) = -Vz • J(z,t)
with
J(z,t) = -DVZ • C (z , t) - QF>Vz • C(z,t)
or
J(z,t)	
-DeffVz 0 C(z,t)
where Deff D + QF> is defined as an "effective" coefficient of diffusion.
The concept of Deff has been used garlier (5) to account for the effect of
convection. In this paper thm notation DS is used for Doff in the solid and
DD for Deff in the melt. With regard to assumption (e), it is customary in
dilute binary systems such as doped semiconductors with small liquidus-solidus
separation to take k to be independent of concentration. This approximation
is in principle erroneous for systems with large ligvi:dus- solidus separation.
I
The concentration dependence of k is known from the phase diagram and it could
i
I
3
be added to the set of boundary conditions (providing that the displacement
rate of the interface is constant - see Appendix). 	 Since no general analytical
treatment of the relationship	 between k(c,t) and R(t) is as yet available, R
and k are taken to be constant,.
2.2	 Development of composition transients associated with directional m_ elting
Making use of the preceding assumptions, Fick's second law, formulated for
a coordinate system which moves with the interface according to the configura-
tion shown in Fig. 1, assumes the form:
a2CS	 aCS	 3C 
US . T + R z-	 at [l]
a-CL 	 3CL	 3CL
DL a---2 + R 8z _ -r (2]
The initial and boundary conditions for equations [1] and [2]• are:
t = 0	 CS = Co	 zelo,+o-] [3]
CL - Co/k	 zc]o,--w] [4]
at the interface (z	 0):
CS =kC
L
	
%Lt?0 [5]
Solute flux conservation through the interface yields:
a C S
	
3C D 	 + R C" - D	 + R C*	 V- t5	 az Iz-O+	 S	 L	 az	 L
> 0	 [6]
Z=0-
Using the change of scales
C=C - Co ["1]
T
TL
Ts
i
A W o	 Co CO/ k	 D
C
^ p :lx PAGE Is
OF POOR QUALnY,
1
Interface
T
I
T"	
r
i
Liquid	 Solid
Distance
i
i
.
E
3
z
r
. 7 -	 ORIGINAL PAae 1,9
OF F'OORp QUALITY
and the dimensionless variables:
R02ty . pz	 [8]	 T n 	 [g]	 a = ps
L	 L	 L
[ 10]
the system of eqs. [1] and [2] becomes:
2E—
S
	 W-Sacsa
	
ay	 aT
aye 
[11]
2;.—
	
ac 	
act
+ -'y	 79T [ 12'ay
with the corresponding initial and boundary conditions:
T s 0	 CS a 0	 y-1 0,4-1 [ 13]
CL = Co (k - 1)	 ye 1 0 ., [14]
at the interface (y = 0);
S = k L + Co (k-1)	 i^-T > 0 [15]
aC	 a	 Is
Ty—y
+ cS = 5-y
y
[16]
y=o+	 y=o-
Y
The set of equal ons [11-12] with the boundary conditions [ 14 to 16] and
the initial conditions [13] can be solved by using the Laplace transforms:
r
YS	 CS e'dT [17]
and
yL =	 FL a
-STdt [18]
t
.
l
8 -
	
t3RtG^i^A' PAGE 19
OF poOR QUALITY
yielding:
2YS dYd	 S,.
a22 } dy - 4 YS = 0
d2YL dYL
dyz + J	 b YL . -q
with
9 = Co (. - 1)
Taking the general solutions:
YS A e-ay/a
YG = B e-by + b
and using [13 to 16], one obtains:
exp( 2 - 4+°r)a
is = e
k- 2a - 2- 2 q+ba - A41+ba-73
and
gexp(-2+ 4 +6)yYL = b [l+
	
	 ]	 [251
k-Za-2- 3 2 1+ba - 4 +b
a
These relationships depend mainly on the two parameters k and a = D S/UL. The
inverse Laplace transforms are obtained in the general case by a numerical
method. Analytical solutions will be given for particular values of a.
The inverse Laplace transform is made by approximating the integral:
[191
[201
[21]
[221
[231
[241
	9 -	 0"914iGINAL PAGE IS
OF POOR QUALITY
c+i«
U (y .T) -
	
	 e4T7(6)d6
	
[26]
c-ice
by a trapezoidal rule using discrete values of the Laplace transform Y(A) along
I
the integration path ,6 a c + iw. For this purpose the integral [26] is approx-
imated by Fourier cosine series which are evaluated using the Fast Fourier
'	 transform algorithm.	 The numerical inversion of (24] gives CS(y,T) for
given values of of and k. Figure 2 is a plot of T0.99, the time necessary to
reach 99% of steady state as a function of k, for four different values of a
obtained by the inversion of [24]. [The real time (t) is related to the dimen
sionless time T through t = DST/R2.]
At the hypothetical upper limit (a = 1), the liquid and the solid behave
symmetrically and a corresponding solution of [24] is given by:
	
CS (y,T)	 2—°(1 - k)erfc(	 2-	 ( -y) - 2— erfc(+--2 )	 [27]
which reduces at the interface to:
S(0,T) - Co (k - 1) erf Z	 [28]
Equation [28j confirms the plotted data, 99% of steady state (erf 2 - 0.99) is
i
reached for all values of k# 1, with 'T = 13.4. The corresponding curve on
Fig. 2 is used as a reference line to determine the variation of T0.99 with ar
for any given value of k.
Considering the special case of a + 0, the inversion of [24] yields:,
	
(y . T )	 C-0 { ( 1 - k) erfc (,. - ) exp (-y) + I erfe (--Y + )S	 2	 3T	 T 2
2-k	 2-k l	 1-k	 2-k
- ( )exp ( 2k - 2)y exp( 2 T)erfc(2 T + 2k r) }	 (29]k
or,10INAL PAGE IS
OF pooR QUALITY
30
26
18
14:
10
6
2'.
_u5	 6
k-^
04` 
101r',U PACc, 1.9	
- 11 -
00 POOR QUALrry
CS(O,T) • T {(1 - k)erfc(T, ,,- werfc (t)
- ( Z— )exp( T)erfc(- T)	 (30]
2.3 Discussion of segregation analysis
The relationships [29] and (30) can be applied to Solidification by using
the transform S defined by k 4 1/k, S —S+ L, L s+ S. It yields a solution
which is identical with that given by Smith et al. (7) where DS is considered
negligibly small. Figure 2 shows that T curves drawn for different slues of
a converge to equation (291 for a 5 10 -4 . Generally accepted values for DS and!
DL in semiconductors lie in the range 10-10 -10 8 cm2/sec for DS and 5x10-4
10-5 cmz/sec for DL 'C"utrefore the Smith et al. solution (or its equivalent for
melting [291) must be considered a good approximation. It should be pointed
out, however, that as yet there are only limited data (of reasonable precision)
for diffusion coefficients at melting point temperatures. While the uncertainty
concerning the magnitude of DL is not expected to have a noticeable effect on
the value of a. DS can be considerably larger than determined by extrapolation
from ',ower temperatures The temperature controlled increase in vacancy con-
centration, for example, will affect DS and may thus have a significant effect
on the value of a. The data show that already for a = 10 -3 the Smith et al.
approximation (or eq. [29]) no longer applies, and that for any given interface
displacement rate, the compositional transients in the solid (T0.99) (and in
the liquid) increase with increasing a for k < 2 and decrease for k > 2.
2.4 Implications of segregation behavior on seeded crystal growth
It is of interest to consider the present analysis in terms of real time
(t) behavior, i.e, to consider, for example, the specific effect of the rate of
12 -
directional molting (R) on the time required to reach steady state. 	 For this
purpose, it is convenient to analyze the behavior of specific systems. 	 An
0 arbitrary choice is 11g l,xCdxTo of the composition x = 0.2 (Co n 0.2).	 Figure
f 3 presents tho concentration of CdTc in the solid (HgCdTe) at the crystal -melt
interface (CS) as a function of T for kS
 and kh .	 In real time (t), the CdTe
concentration (CS) is dependent on the rate of melting (R) which, in order to
be meaningful, must satisfy the stability criteria (3). 	 At the beginning of
melting, the system is at equilibrium CCS(z,0) W Co, CL (z.0 , a C0 /kJ, and
therefore under conditions of absolute'stability.	 As	 melting proceeds, the
interface stability is affected by out-diffusion and the subsequent formation
of the boundary layer in the solid. 	 Thus, while For Ch (O,z)	 Co/k melting
can be initiated at any rate, out -diffusion and interface breakdown must be
anticipated if the rate exceeds a critical value.
	
If the stability criterion
is strictly applied to a steady state-melting configuration, the GS/R ratio
required for stability is very high (~10 8OC/sec for Hs0.8Cd0.2Te).	 Such a high
value is achieved by either a very high thermal gradient or a very slow rate
of interface displacement. 	 Making use of Fig. 3 and applying numerical values
for HgCdTe, th-, present analysis shows that under neither condition will
steady state be reached	 during seeding in Bridgman or Czochralski
growth	 whore the seed 3s back-melted for several millimeters only (see
Table 1); the =Ysta.l-melt interface temperature during molting must therefore be
k,
expected to lie somewhere between the solidus and the liqudus temperatures.
Considering the back-molting of an ingot of fig MCd0.2Te for half ai hour at
the rate of 3,7 x 10 -2 pm/sec (the rate required for absolute stability at
steady state if GS
 = 1000°C/cm), the T value at the end of back-melting (i.e.
after half an hour) is 4 x 10 -4
 and, according to Fig. 3, the concentration in
the solid (CS) will be essentially unchanged at C o
 and the interface temperature
will remain at about TS,
ORIGINAL PAU 1.9
OF POOR Q()ALny
00
loW.C>
A o
E
C"%i crCs
Lr;Mt
C) C> C)	 C)	 C> 	 ^'^ _^' aC3 CD	 Cr2
C"
3
(0 0) ainlDladwal
•
4 o
op 0
00
000 0 0
400 doO
00
0
rn
C', I t-li
1-4
C)2
i
O
-' 14 -
The dependence of the length of transient behavior on the magnitude of
the liquidus-solidus separation can be demonstrated by comparing Hg0.8Cd0.2Te
with Ga-doped Ge where the liquidus-solidus separation is very
small (less than l°C). The T curves of Ga-doped Ge are presented in Fig. 4.
At the composition 10 18
 at/cm3 of Ga. the GS/R ratio assumes the value of
2.9 x 103 °C/sec (MS = 0.3660C/%; DS - 10-8 cm2 /sec; k - 0.087). Since the
Ga-doped Ge system has been extensively studied, the credibility of the data
used for the different parameters is much higher than for HgCdTe. Table I
shows that the length to be melted to reach steady state (xm) in doped Ge is
much smaller than in Hg0.8Cdo. 2Te. The interface concentration in the solid
under the conditions depicted in Table I can be calculated using the T curves
in Fig. 4. Assuming again that back-melting is to proceed for 0.5 hours at
GS = 10°C/cm and R = 34 um/sec, we obtain a r value of 99,which indicates
that steady state has been reached and that the concentration in the solid
at the interface, CS,,is kCo. For this system, T0.99 (steady state) is already
reached after about 9 minutes. With continuing melting, the crystal-melt
interface temperature changed from T S to TL , the liquidus temperature. It is
noteworthy that for systems with a liquidus-solidus separation smaller than
in Hg0:8Cd0.2Te but larger than in very dilute systems, the temperature of the
interface (T *) at the end of back-melting is expected to be somewhere
between TS and 1' L; its value, at a given time, can be calculated from
appropriate T curves. The interface temperature is related to the
composition at the interface through the relation T* m + m tCS, where Tm is
the melting temperature of the major constituent (for example, HgTe in
1190.8Cd0. 2Te) and mt is the slope of the straight line joining Tm and the point
on the solidus curve corresponding to the concentration CS on the phase
diagram. Therefore, the deviation of the interface temperature from the
4
ORIGINAL PAGE M
OF POOR QUALITY
•
	
C o	 ^.
	
°' a	 Ga doped Ge
.	 n 
t	 ^{	 M = O,OII7
{	 •
,
r*
VS
{
1{
1
Q ^
	0 	 10 X20	 30	 40	 50	 60
T --
16 -	 ORIGINAL PAC9 13
OF POOR QUALITY
:solidus temperature during transient melting (AT * a T* -TS) in non-dimensional
notation is given by:
^T* mtCS mt=0Co	 (31)
3	 As C* approaches the value of W p AT E approaches a maximum value for
which T* - TL' `	 k	 % ,
The preceding analysis and conclusions apply to initial conditions whereby
a (binary) solid is in thermodynamic equilibrium with the contacting malt
(CS * kCd . This condition can in prin(% p'le be realized during seeding in
Czochralski and Bridgman-typo growth. The analysis as such, however, is not
applicable to configurations in which initially the entire charge is solid and
seeded "re-growth" is accomplished by first directionally melting it ,fraction
of the solid [configuration used during growth experiments in reduced gravity
onvi.ronment( g)). Initial melting in this case cannot a priori be expected to
take place at TS . If the phase transformation is initiated under diffusionless
conditions, melting will take place at T S , To < T el , a temperature at which the
solid and the liquid ferr2d have equal free energies (J) but are at a different
chemical potential. Maintaining this kinetically controlled metastable state
will obviously be a function of the experimental conditions under which melting
proceods. Depending an the extent of di£fusional adjustment of tile phase
boundary composition toward equilibrium, a transient in the crystal-melt inter-
face temperature will be encountered. Thee transients to equilibrium for
systems with CG CSjk are currently under investigation.
APPENDIX
As a consequence of segregation during transient me lting or so lidifi cation
the grow th-interface temperaturo (1'*) is subject to change and the interface
a
3
i.	 /f ..,	 .	 ... ,..	 ,.	 »..... _.._: ...	 ........+r...,wr 	 w..wrw+.awywawrr.-w....,..


Segregation during directional melting and its implications on seeded crystal growth: A theoretical analysis

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