Optimal zone lengths in multi-pass zone-refining processes

ELSEVIER

Separations Technology 6 (1996) 227-233

Optimal zone lengths in multi-pass zone-refining processes Chii-Dong Ho*, Ho-Ming Yeh, Tzuoo-Lun Yeh Department of Chemical Engineering, Tamkang University Tamsui, Taiwan, R O.C.

Received 16 April 1996; accepted 1 June 1996

Abstract A method for determining the optimum zone length for maximum separation in multi-pass zone-refining processes is described. Numerical values of the optimum zone lengths and their corresponding maximum separations for up to ten passes are presented. The optimum zone length is found to increase with the distribution coefficient but decreases when the pass number increases. Considerable improvement in separation may be obtained if zone refining is operated at the optimum zone length, except for higher distribution coefficients. Keywords:

Zone refining; Multi-pass; Optimum zone length

1. Introduction The general

Heater w--+

term zone refining denotes a family of

methods for controlling the impurities or solute concentrations in crystalline materials. Zone refining is an extremely efficient puriiication technique for separating liquid or solid mixture [1,2]. The first important application of zone refining was used to purify germanium in transistors [31. In multi-pass zone refining, a series of closely spaced heaters is placed in relative motion to a long solid ingot leading to a redistriiution of solutes in the ingot, as shown in Fig. 1. The advantage of multi-pass zone refining is the savings in operating time, as succeeding crystallization begins before the preceding one is completed. With proper selections of the zone number and zone length, many useful operations can be performed with solid ingots. Birman has designed a matrix method to describe the zone refining of a bar with non-uniform initial solute concentrations [4]. Zone refining with zone length changing along the ingot has been studied [5-81. Davies found in his studies that the optimum ratios of zone length to ingot length are 1.0 and 0.3, respectively, for the first and second pass [51. Jackson and Harm discovered a method for producing a linear gradient of solute concentration by changing the 0956-9618/96/$15.00 0 1996 Elsevier Science Ireland Ltd. All rights reserved. PII SO956-9618(96)00160-9

l-u X

\

Moltenzone

c

(a) Single-pass wne refining

Ring heater ( Stationary )

Molten zone

@) Muti-pass zone refining Fig. 1. Schematic diagram of zone refining apparatus.

C.-D. Ho et at! /Sepamtions

228

volume of a molten zone [a]. Wilcox developed a theory for the determination of the non-constant distribution coefficient, and Yeh and Yeh developed the separation theory and investigated the improvement of separation in a two-pass zone refining process with zone length varied along the ingot [7,81. A large body ‘ofsign&ant investigations devoted to some aspect of applications in zone refining have appeared in the literature during the last two decades. For instance, the silicon-on-insulator (SOI) films [g-12] and semiconducting [13-B] and superconducting [16-181 substances were prepared by zone-refining processes. The purpose of this work is to determine the optimum zone length for maximum solute removal in multi-pass zone-relining processes and investigate the improvement in separation which may be obtained when the operation is performed at the optimum zone length. 2. The basic equations

Technology 6 (1996) 227-233

zone length is no longer constant, and solute is no longer taken in. The equation becomes:

= -c,wdx,

d[U -x)C,W/kl

(L-OSXSL (2)

or K,(x)

= ~C”(X)dr,

(L-OSXSL

(3)

Note that above equations satisfy the boundary conditions:

at x=0,

C” = (k/Z$C”_

at x=L-1,

,(xMx

(4)

LC, - i’-‘C,,(x,dx]

The basic differential equation applicable to multi-

(5)

pass II zone-relining process with same zone length 1 in each pass was derived independently by Lord [19] and Reiss [20]: Id[C,W/kl

= [C”_,(X+ 0 - C”(XMx, OSXSL-1

(1) k -0.5

with the meaning of the symbol given in the Notation

section. . This equation is obtained by a mass balance of the solute within the moving zone ABCD or A B’C’D’as shown in Fig. 2, and based on the following assumptions: (a) constant distribution coefficient; (b) uniform composition in the liquid, (c) negligible diffusion in solid; (d) constant zone cross-section; (e) no volume change on freezing; (0 constant length of liquid zone. After the front of the zone reaches the end of the ingot, at x = L - 1, where L denotes ingot length, the

&

1.0

molten zone AEKD or AWCI)’

y’ .,I*-..*..-**... 0.0

0.2

. *. . . . *. . . 0.4

0.6

0.8

1.0

X

Fig. 2. Schematic diagram of zone refining operation.

Fig. 3. The effect of zone length on the concentration distribution of zone refming after two-pass operation (k = 0.5).

C.-D. Ho et al. /Sepamtions

The solutions of Eqs. (6) and (7) associated with the boundaxy conditions, Eqs. (8) and (91, are:

Setting 4,(X)

229

Technology 6 (1996) 227-233

= C,(X)/C, X=x/L Y=I/L

Eqs. (1) and (3) can be rewritten

&J”(X)

&%(X)=

= [A_ ,(X+

=C#J 1-X

+

as:

O~X~l-Y

(6)

l-YSXI;l

(7)

n(XMX,

= k(1

4”(X)

Y) - 4JX)lG

I

/oy4n_Io~, -xjk- ‘F,(Y),

4” = ($)[l-

l-Y
(11)

1

I;,(Y) =Y-k 1- $-y+JtMC

atX=l-Y,

(10)

where

1

and the boundary conditions are at X= 0,

OI;XI;l-Y

(12)

(8)

2.1. One-pass opemtions

(9)

For one-pass operation, it = 1, and 9&X + Y> = 4,(X) = 1. Thus, the concentration distribution in the ingot after one pass is obtained from Eqs. (lo-121

Jy-y4”(X)dx]

‘O’ 9 O

1 k =OS

01 / 0.0

0.2

0.4

06

0.8

IO

X Fig, 4. The effect of zone length on the concentration of zone refining after two-pas operation (k = 1.5).

distribution

Fig. 5. The effect of zone length on the concentration of zone refining after five-pass operation (k = 0.5).

distribution

C.-D. Ho et al. /Sepamtions Technology6 (1996) 227-233

230

as

C&(X) = k(1 -XIk= 1 - (1 - k)e-‘k’Y’x

&(X>

Os;X
,

‘F,(Y),

1 -Y
1 (17)

where (13)

&(X)

=k(l

1 - YzzX<

-Xjk-‘F,(Y>,

F,(Y) = (1 + [l - e-k(‘-Y)/Y](l

1

(14)

(18)

(15)

The results for k = 0.5 and 1.5 are presented graphically in Figs. 3 and 4, respectively. For further-pass operations, the concentratibn distributions are readily obtained if we perform the calculations of Eqs. (lo-121 repetitively. Figs. 5 and 6 show the concentration distributions of five-pass operation for k = 0.5 and 1.5, respectively.

- k)/k)/Yk-’

2.2. Multi-passoperations For two-pass operation, n = 2 and the concentration distribution may be calculated numerically from the following equations with the use of Eqs. (13-15)

3. Optimum zone length

OI;X<

+

1 -Y

(16)

Since the initial dimensionless concentration d,,(X) is equal to unity, the solute removal can be calculated

0.5

‘/O’O

0.1 0.0

0.2

0.4

0.6

0.8

1.0

X

Fig. 6. The effect of zone length on the concentration of zone refining after five-pass operation (k = 1.5).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

I

Y

distribution

Fig. 7. The effect of zone length on solute removal after two-pass operation UC< 1).

C.-D. Ho et al. /Separations

231

Technology 6 (1996) 227-233

Mathematically, the optimum zone length Y* for maximum W, is obtained by solving the equation, dW, /dY= 0, with the use of Eqs. (13-15). Since (dW, /dY) < 0 for k < 1 and (dW,/dY) > 0 for k > 1. In order to meet both situations, therefore, Y must be as large as possible, i.e. normal freezing for all values of the distribution coefficient. This is the same result obtained by Davis [5]. Accordingly, the optimum zone length for the first pass is

\

II

I*=LorY*=l

k=1.7

(21)

and the concentration distribution is readily obtained from Eqs. (14) and (15) by setting Y = Y* = 1 4;(X)

05x5

= k(1 -Xjk-‘,

1

(22)

while the maximum solute removal W, max is calculated by substituting Eq. (22) into Eq. (2b). The result is shown in Fig. 10 for n = 1. 3.2. Multi-pass operations The optimum zone length Y* for ma$mum H(, after multi-pass operations can be obtained from Eq.

k =1.1

045

o17.

’

01

0

.

’

*

02

03

.

’

04

.

*

0.5

‘.

’

06

07

.

’

’

08

0.9

1 n=2

1

Y 04

Fig. 8. The effect of zone length on solute removal after two-pass operation (k > 1).

bY 0 35

/,.j, 1 - &,(X)ldx

= 0.5 - iO%#JJX)dX,

0

w, =

for k < 1 /O.5[ +JX)

- I]&=

~“~5~n(xMx-

03

0.5,

0

0

fork>

1 (19)

The optimum zone length Y* for maximum separation may be obtained by partially differentiating W, with respect to Y and solving Y* from the equation dW,/dY=

n=4

‘>

0 25

n=5

:

0;

n=v

=I

0.

n=9

n=IO

3.1. One-pass operations

0 I!

For one-pass operation, Eq. (19) becomes fork<

0.5 - ~“.s$,(x)dx,

01

1

00

02

04

06

08

(20)

w, = /

““&(x1&-

0.5,

for k > 1

10

12

14

16

18

2.0

k

Fig. 9. The optimal processes.

zone

length

of multi-pass

zone

refining

232

C.-D. Ho et al. /Sepamtions

Technology 6 (1996) 227-233

4. The improvement in separation

The improvement in separation by operating at the optimum zone length Y* is best illustrated by calculating the percentage increase in separation based on that operating at Y

Ill=

Wn ,max w

-

n

wnx 100%

(23)

For illustration, the results for n = 2 and 5 were calculated by Eqs. (19) and (23) with the use of Figs. 3-6 and are given in Tables 1 and 2.

5. Results and discussions 0 I5

0.1

005

lit.........

0 00

0.2

0.4

0.6

08

1.0

1.2

1.4

1.6

1.8

2.0

k Fig. 10. The maximum solute removal of multi-pass zone refining processes.

(19) numerically with c$” (X1 calculated previously from Eqs. (10-12). For illustration, Figs. 7 and 8 are the graphical representations for the effect of Y on W,, as well as for determination of Y* with two-pass operation for k < 1 and k > 1, respectively. An alternate representation for Y* with two- and five-pass operations are also shown in Figs. 3-6. The results up to ten-pass operations are given in Figs. 9 and 10 for Y* and W,. max,respectively.

The effect of zone length on the separation efficiency in multi-pass zone-refining processes has been investigated with the distribution coefficient as parameter. The concentration distributions &, after nth-pass operation were calculated from Eqs. (10-12) numerically. Figs. 3-6 illustrate the concentration distributions along the ingot for two- and five-pass operations. The solute removals W, were calculated from Eq. (19) and the optimum zone length Y* for maximum solute removal W,, max were determined by the condition: dW,/dY= 0 at Y = Y*. Figs. 7 and 8 show the existence of optimum zone length with two-pass operation as example. The existence of Y* with twoand five-pass operations are also shown in Figs. 3-6. The results up to ten-pass operations are given in Figs. 9 and 10 for Y* and W,,,,,, respectively. It is shown in Figs. 9 and 10 that the optimum zone length increases with the distribution coefficient but decreases when the pass number increases while the maximum solute removal increases as the distribution coefficient goes away from unity at which no solute is

Table 1 Improvement in separation for two-passes operations Y= 0.2

Y= 0.4

k

Y*

w2mx

w,

I2

w2

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 2.0

0.421 0.424 0.427 0.428 0.430 0.433 0.433 0.434 0.435 0.438

0.4860 0.4019 0.2852 0.1639 0.0513 0.0475 0.1314 0.2011 0.2583 0.3248

0.4792 0.3700 0.2420 0.1287 0.0376 0.0330 0.0871 0.1289 0.1614 0.1873

1.419 8.622 17.851 27.350 36.436 43.939 50.861 56.012 60.037 73.412

0.4860 0.4016 0.2847 0.1634 0.0511 0.0473 0.1306 0.1997 0.2564 0.3026

Y=O.6 -

Y=O.8

12

w,

12

w,

12

0.000 0.075 0.176 0.306 0.391 0.423 0.613 0.701 0.741 7.336

0.4709 0.3736 0.2589 0.1470 0.0457 0.0424 0.1175 0.1808 0.2338 0.2779

3.207 7.575 10.158 11.497 12.254 12.028 11.830 11.228 10.479 16.877

0.4487 0.3354 0.2250 0.1256 0.0388 0.0360 0.1002 0.1553 0.2026 0.2432

8.313 19.827 26.756 30.494 32.216 31.944 31.138 29.491 27.493 33.553

C.-D. Ho et al. /Separations

Technology 6 (1996) 227-233

233

Table 2 Improvement in separation for five-passes operation Y = 0.2

Y=O.4

Y = 0.6

Y= 0.8

k

Y*

W5,max

K

15

W5

15

W5

1s

4

's

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 2.0

0.188 0.215 0.229 0.237 0.243 0.247 0.250 0.251 0.252 0.254

0.4998 0.4801 0.3952 0.2490 0.0809 0.0749 0.2019 0.2972 0.3651 0.4288

0.4998 0.4799 0.3929 0.2449 0.0786 0.0722 0.1936 0.2848 0.3507 0.3972

0.000 0.042 0.585 1.674 2.926 3.740 4.287 4.354 4.106 7.956

0.4953 0.4479 0.3458 0.2095 0.0669 0.0620 0.1689 0.2528 0.3165 0.3640

0.909 7.189 14.286 18.854 20.927 20.806 19.538 17.563 15.355 17.802

0.4733 0.3827 0.2687 0.1533 0.0477 0.0440 0.1214 0.1858 0.2391 0.2830

5.599 25.451 47.079 62.427 69.602 70.227 66.310 59.957 52.698 51.519

0.4491 0.3362 0.2257 0.1260 0.0389 0.0361 0.1004 0.1554 0.2027 0.2433

11.289 42.802 75.100 97.619 107.969 107.479 101.096 91.248 80.118 76.243

removable. The improvement in separation by operating at the optimum zone length is illustrated for twoand five-pass operations, as shown in Tables 1 and 2, respectively. it was found in these tables that the improvement Z,, increases with the distribution coefficient. Notation

c

solute concentration in the freezing solid at solid-liquid interface, (g/cm3) C,(x) solute concentration freezing out of the zone at distance x in the nth pass, (g/cm3> solute concentration in the main body of the liquid zone, (g/cm3) d%(X) dimensionless concentration C,,(x)/C, uniform solute concentration, (g/cm3) C” the percentage of separation 1” improvement defined by Eq. (23) k distribution coefficient defined by C/C, L ingot length, cm 1 zone length, cm separation defined by Eq. (19) Wn,max W, obtained in the process with optimal zone length x distance from the starting end of the ingot, cm Y dimensionless zone length, Z/L Y* optimal value of Y X X/L

w,

References [II Pfann, W.G. (1964) Zone Melting, 2nd ed. New York: John Wiley.

Dl Lawson, W.D. and Nielsen, S., (1962) New Chemical Engineering Separation Techniques. In: Schoen, H.M. (Ed.), New York: John Wiley-interscience. 131 Pfann, W.G. (1952) Principles of zone-melting. Trans. AIME 194, 747. [41 Birman, J.L. (1953) On zone refining. J. Appl. Phys. 26, 1195. [51 Davies, L.W. (1959) The efficiency of zone-refining processes. Trans. AIME 215, 672.

[61 Jackson, K.A. and Pfann, W.G. (1965) Producing a linear gradient of solute concentration by changing the volume of molten zone. J. Appl. Phys. 36, 320. 171 Wilcox, W.R. (1978) Non-constant distribution coefficients from experimental data: theoretical. Mater. Res. Bull. 13, 287. P31 Yeh, H.M. and Yeh, W.H. (1979) The improving of separation in zone refining processes with zone length varied along the ingot. Sep. Sci. Technol. 14, 795. [91 Fan, J.C., Geis, M.W. and Tsaur, B.Y. (1981) Lateral expitaxy by seeded solidification for growth of single-crystal films on insulators. Appl. Phys. Lett. 38(51, 365. no1 Kubota, K., Hunt, C.E. and Frey, J. (1985) Thermal profiles during recrystallization of silicon on insulator with scanning incoherent light line sources. Appl. Phys. Lett. 46(12), 1153. [ill Miaouli, I.N., Wang, P.Y., Yoon, S.M., Robinson, R.D. and Hess, C.K. (1992) Thermal analysis of zone-melting recrystallization of silicon-on-insulator structures with an infra-red heat sources: an overview. J. Electrochem. Sot. 139(91, 2687. [121 Richer, H.H. and Tillack, B. (19931 Thermal analysis on zone-melting recrystallization of silicon-on-insulator films using a finite difference method. Mater. Sci. Eng. A173, 59. [131 Cohen, S.S., Wyatt, P.W., Chapman, G.H. and Canter, J.M. (1988) Laser-induced diode linking for water-scale integration. IEEE Trans. Electron Dev. 35(9), 1533. [141 Lord, H.A. (1988) Thermal and stress analysis of semiconductor wafers in a rapid thermal processing oven. IEEE Trans. Semicond. Manuf. 1 (3), 105. u51 Young, G.W. and Chait, A. (19891 Steady-state thermal-solutal diffusion in a float zone. J. Crys. Growth 96 (11, 65. [I61 Mani, T.V., Damodaran, A.D. and Warrier, K.G.K. (1994) Zone melting-refining in Bi-Pb-Sr-Ca-Cu-0 superconductor prepared through the Sol-Gel method. J. Mater. Sci. 29 (24). 6453. [171 Schaper, W., Bormann, R. and Freyhardt, H.C. (1993) Laser processing of Nb//3Al(Ge,Si) high-field superconductors. J. Appl. Phys. 74 (41, 2686. [181 Warrier, K.G.K., Varma, H.K., Mani T.V., Damodaran, A.D. and Balachandran, U. (1995) Zone refining of sintered, microwave derived TBCO superconductors. J. Mater. Sci. 30(12), 3238. 1191 Lord, N.W. (1953) Analysis of molten-zone refining, Trans. AIME 197, 1531. DO1 Reiss, H. (1964) Mathematical methods for zone-melting processes, Trans. AIME 200, 1053.