Numerical simulation of single crystal growth by submerged heater method

Journal of Crystal Growth 104 (1990) 233—238 North-Holland

233

NUMERICAL SIMULATION OF SINGLE CRYSTAL GROWTH BY SUBMERGED HEATER METHOD Aleksandar G. OSTROGORSKY Mechanical Engineering Department, Columbia University, New York, New York 10027, USA Received 27 October 1989; manuscript received in final form 2 March 1990

A novel method of crystal growth which utilizes an axial submerged heater is proposed and numerically simulated. Single crystals should be grown by directional solidification in vertical bottom seeded crucibles. Submerged in the melt, the heater supplies the heat axially downward, enclosing and stratifying a small active portion of the melt.

1. Introduction In all reported studies of crystal growth by the vertical bridgman (YB) method, the solidification process was carried on inside of tube type furnaces [1]. Muller [2] reviewed the reported results and summarizes the established advantages and disadvantages of this method. In tube type furnaces the heat is supplied radially inwards, through the crucible wall, inducing radial temperature gradients in the growing crystal and the melt. It is established that radial temperature gradients: promote longitudinal macroscopic segregation, caused by convective mixing in the melt [3]; result in a nonplane, usually concave solid— liquid interface [3,4]; induce thermal stresses and dislocations in the grown crystal [5], and upon exceeding a critical value; drive the unsteady natural convection in the melt causing microscopic inhomogeneities in the grown crystal [3]; To ameliorate the above adverse effects, a number of computer controlled multisegment tube type heaters were developed [6]. Muller [2,7] proposed the use of conductive graphite plugs to enhance axial heat transfer in the melt and the growing crystal. In this communication, a novel method of directional solidification is proposed and numeri—

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0022-0248/90/$03.50 © 1990

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cally simulated. The basic features of this method will be presented here, while the details of the apparatus are described in ref. [8].

2. Description of the proposed method A schematic diagram of the method is shown in fig. 1. The melt is contained in a bottom seeded crucible. The heat is supplied to the melt axially by the main disc-shaped heater submerged inside the melt. During the growth process, the heater is held at a temperature TSH somewhat higher than the melting point of the charge, Tm. The crucible is insulated radially. The contact between the heater and the melt is prevented by an inert jacket made preferably out of the same material as the crucible. A highly conductive cylindrical plate placed between the heater and the melt further improves the uniformity and symmetry of the axial heat flux imposed on the melt. A tube type guarding heater is placed around the enclosed melt to control the radial heat flow. The heat is lost mainly in the downward direction through the crystal. The heat is extracted from the growing crystal through the crucible support attached to the shaft. The crystal is grown by lowering the crucible. The heaters and the insulation remain at a fixed position. While the crucible is lowered, the melt

Elsevier Science Publishers B.V. (North-Holland)

234

AG. Ostrogorsky / Numerical simulation of single crystal growth by submerged heater method

_________fl_________

3. Simulation of fluid flow and heat transfer

10

~ ~~IL~&~i

The spectral element program NEKTON [9] was used to simulate the heat transfer and fluid flow in the proposed process. Axial symmetry is assumed. The hot zone of the apparatus was divided into 69 large spectral elements (fig. 2). The hot zone consists of four materials: gallium arsenide (GaAs) melt, GaAs single crystal, silica crucible and crucible support and alumina insulation. The velocity, temperature and pressure within the spectral elements were expanded as 5th order tensor product Lagrangian interpolants through Chebyshev collocation points.

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~

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6

7

Tm ~TAux

____

8

____

Fig. 1. Schematic of directional solidification by the submerged heater method [8]: 1, melt; 2, crucible; 3, heater; 4, inert jacket; 5, conductive disc; 6, guarding heater; 7, support member; 8, shaft; 9, crystal; 10, auxiliary heater; 11, insulation.

Insulate~-ø~1 Melt

~___-h

I Tm

solidifies at a small constant distance H below the bottom surface of the submerged heater. Assum ming a constant axial temperature gradient aT/az in the submerged melt, the distance H is equal to

+~

\

H

H= 8T/Bz’

where

(1)

eff’ T,,,,, -

~1

~

~

+ ATGH

-______________ ________________________

GaAs ~‘SH

I

_____

hen,T,,,,

______ _____ _____________

L~T~H = T

5H Tm. Note that for a constant heater temperature (i.e. constant zITSH), H depends only on the axial temperature gradient aT/az in the enclosed melt. Changes in 3T/8z during growth are mild and can be compensated by changing L~T5~. An auxiliary heater is located above the melt to provides means for temperature control in the upper portion of the crucible. This heater can be used to maintain the charge in molten state and to thermally stratify the melt above the main heater. —

Alumina ~5~ICa

h efi

5 cm

~

Fig. 2. Geometry of the hot zone, spectral elements and boundary conditions (crystal diameter =10 cm; crucible thickness 0.25 cm; insulation thickness = 1.75 cm).

/ Numerical simulation of single crystalgrowth

A. G. Ostrogorsky

Fluid flow and heat transfer in the melt are coupled to the heat transfer in the hot zone. Therefore, the continuity, Navier—Stokes and energy equation governing the time-dependent viscous flow in the melt [10], were solved simultaneously with the transient energy equation in the hot zone. By modeling heat transfer in the entire hot zone, realistic thermal boundary conditions are imposed on the melt boundaries. Fluid boundary conditions: No-slip was assumed on all boundaries, Thermal boundary conditions are shown in fig. 2. It was assumed that each heater imposes a constant uniform temperature. An effective heat transfer coefficient hell was used to account for the convective and radiative heat losses from the surface of the insulating cylinder, —

a ar k1~,T/

— —

‘~T T ~

hell t.

S

‘~ ~

—

where k~05is the thermal conductivity of the insulation, T~is the surface temperature and h elf is defined as heff = 4a(

T+T 2

by submerged heater method

235

where a is the Stefan—Boltzmann constant and h is the convective heat transfer coefficient. Radiative heat losses, depend strongly on 7~. Since T, is not known, the value of h elf was determined by iterative simulations. At each iteration, a new more accurate surface temperature was obtained and used in eq. (2) to compute heff. The iterative simulations indicated that the temperature field in the hot zone is only weakly affected by changes in hCff; the insulation presents the dominant resistance to the radial heat flow. Most of the heat is conducted axially downward through the growing crystal. The boundary condition at the bottom of the hot zone is, ~

aT/az

=

helf(Ts

—

Tj,

(3)

where ~ is the conductivity of the support member and 7~is the temperature of it’s bottom surface. Initial conditions: Conduction steady state solution in the hot zone and the melt was used as the initial condition for the transient simulation.

~ +

~

(2) 4. Results and discussion —

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The results are given in the dimensional form to enable easy companson with the growth condi tions in conventional YB systems To determine the axial temperature gradient in the enclosed melt the temperature of the sub merged heater TSH was varied between 10 and 20 K above T Without the use of the guarding heater the axial temperature gradient in the en closed melt was in found to be aT/az 18 K/cm of magrntude A change of 10 T~H K inis two the orders temperature of the submerged heater smaller than the dnving potential for the axial downward flow of heat, ~ ~ 1000 ~ Therefore, variations in ~ caused only 1% change in the axial heat flow rate, resulting 1% change in aT/az. In order to keep the melt interface 1 cm. below the heater (an arbitrarily selected value) the temperature of the submerged heater was set to ~TSH = 18 K above the solidification point of GaAs. =

.~-

-

Submerged heater

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f

,....,‘~...

“~

I

4.—.

-

..

-

Fig. 3. Velocity field in the melt (without the guarding heater). Um~= 0.119 cm/s.

236

A.G. Oslrogorsky

_______

/ Numerical simulation of single crystal growth by submerged heater method —

~i 570 560

1—1550

sary since the solid and the liquid phase have different thermal properties. The curvature parameter of the melt—crystal interface defined in ref. [3]

~—

I— 1540 I L

r

1529 ~l 529 1511

Growth

______________

interface

~

1 43~

iterative computations: the spectral element mesh was deformed to follow the isotherm Tm = 1511 K separating the phases. This procedure was neces-

and ~z/D is approximately equal to 0.03 (~z is the deflection and D is the crystal diameter. Due to the axial heat supply, the vertical temperature difference which opposes the flow in the enclosed melt, ~ ~~ is much larger than the radial temperature difference T~ 2 K, which drives the flow: ~

=

~

1 38b

~

1330~ 1 28b

=

~~ai/~9. ~..—

The Rayleigh number based on ~ 7. is equal to 4 1050. Rar g~(~T~/H)H =

l23~~

_______________

Fig. 4. Temperature field (K) in the hot zone (without the guarding heater).

The submerged heater stratifies the enclosed melt, while at the same time imposes Bernard-type boundary conditions on the top melt. To suppress unsteady natural convection in the top melt (and simplify our numerical simulation), we stratified the top melt by keeping auxiliary heater at TAUXH 68 K above the solidification point (50 K higher than TSH). Figs. 3 and 4 show the velocity and the temperature fields that occur in the absence of the guarding heater. The flow in the melt is steady. The highest velocity achieved in the melt is 0.12 cm/s corresponding to a Reynolds number based on the height of the enclosed melt,

=

Note that if z~TSHis reduced by a factor of 2, H will be also reduced by a factor of 2, reducing the Rayleigh number by a factor of 2~ 16. The growth conditions are further improved when a tube type guarding heater is placed around the enclosed melt [8]. Fig. 5 shows the velocity field in the presence of a guarding heater which is =

-

.—-

7~

‘

I

-

~

=

Re=UmaH/v=24. This is almost two orders of magnitude lower than the Reynolds number in a 5 cm diameter YB melt [11]. Without the use of the guarding heater, all isotherms are concave because the heat flow radially outward. The concave shape of the growth interface shown in Figs. 3 and 4 was obtained by

,

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t

-

1

I

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~.

I

Submergedheater -

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I Fig. 5. Velocity field in the melt (with the guarding heater).

U,~ 5= 0.064 cm/s.

A.G. Ostrogorsky

/ Numerical simulation ofsingle crystal growth by submerged heater method 01.

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1520

237

.......~ , __________

.

0.5 cm 1cm

.

iii _______________

~111111111111111111111111 IlIlillIllIl

II submerged heater II b)

1500 1480 a

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5cm

z

1340 0

1

2

3

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[cm] Fig. 6. Temperature field (K) (with the guarding heater) as a function of radial position and distance away from the submerged heater. Enclosed melt, distance = 0.5 cm; crystal, distance=1 to5cm.

kept at the same temperature as the submerged heater (TGH = TSH). Fig. 6 shows the temperature field in the enclosed melt and the crystal as a function of radial position and distance away from the submerged heater. The guarding heater substantially reduced the radial temperature gradient in the enclosed melt, while the axial temperature gradient changed from 18 to 16 K/cm The ternperature at the melt periphery is less than 1 K above the center line temperature, yielding Ra r 500. The ratio between the opposing and driving temperature difference is increased,

Fig. 7. Radial velocity in the enclosed melt, 2.5 cm away from

the center line: (a) without the guarding heater Umax = 0.0676 cm/s (Re = 14); (b) with the guarding heater Umax 0.0106 cm/s (Re = 2.2).

region surrounded by the guarding heater aT/ar is positive, indicating that the guarding heater is forcing the heat to flow radially inwards. This region extends from the growth interface to 4 cm below the submerged heater. As a result, isotherms are convex. 5 cm below the submerged heater, aT/ar < 0, suggesting that the heat is flowing

6 3 cm

4 cm

4

2 cm

i~T~/M> 16. As a result, velocities in the enclosed melt are

further is almost to Re unaffected 2.2.tothat The Umax by velocity the 0.0106 field cm/s in of the correspondthetop guardmelt ing heater due its low elevation. Fig. 7reduced: shows flow inpresence the enclosed changed direction due to the presencemelt of has the =

=

guarding heater. This implies that the temperature at the melt periphery is indeed higher than at the temperature at center line. Fig. 8 shows the temperature gradients in the crystal as a function of radial position and distance away from the submerged heater. Within the

0’ -2

_______________

01

5cm

2345 r [cm]

Fig. 8. Temperature gradients in the crystal as a function of

radial position and distance away rom the submerged heater: aT/ar > 0, the heat flows radially inwards; aT/ar < 0, the heat flows radially outwards.

238

A.G. Ostrogorsky

/ Numerical simulation ofsingle crystal growth by submerged heater method

radially outward. This region is not affected by the presence of the guarding heater, Radial temperature gradients in the growing crystal can be reduced by the use of a multizone guarding heater. The temperature of the heating elements should be adjusted to follow the temperature change along the growing crystal.

5. Conclusions The submerged heater method yields a very favorable thermal environment for crystal growth, especially in the presence of the guarding heater. The conditions are favorable because the heat is supplied to the melt axially downward, from a very short distance from the growth interface. The submerged heater stratifies the enclosed melt while at the same time acting as a second crucible. The numerical simulations indicate that: the flow in the melt is steady; the radial temperature gradients in the melt and in the crystal are substantially reduced; the shape of the growth interface can be controlled by varying the temperature of the guarding heater; the process is easy to control: all three heaters can be kept at a constant temperature durmg growth. The main disadvantage of the proposed method — —

—

—

is the additional melt contamination from the jacket of the submerged heater.

Acknowledgements The author wishes to thank Professor B.B. Mikic of MIT for his interest in this work, valuable suggestions and for reading the manuscript. Professor T. Patera of MIT has developed the Nekton code used in this simulation.

References [1] W.P. Bridgman, Proc. Am. Acad. Arts Sci. 60 (1925) 305. [2] G. MUller, in: Crystals, Vol. 12 (Springer, Berlin, 1988). [3] D.E. Holmes and H.C. Gatos, J. Electrochem. Soc. 128 (1981) 429. [4] CA. Wang, A.F. Witt and JR. Carruthers, J. Crystal Growth 66 (1984) 299. [5] AS. Jordan, A.R. Von Neida and R. Caruso, J. Crystal Growth 76 (1986) 243. [6] E.M. Monberg, P.M. Bridenbaugh, H. Brown and R.L. Barns, J. Electron. Mater. 4 (1989) 549. [7] G. MUller, Deutsches Patentamt DE 3441 707 Al (1986). [8] A.G. Ostrogorsky, US Patent Application S.N. 397,741 (1989). [9] Nekton Users Manual, Nektonic (1987). [10] A.V. Tangborn, Doctoral Thesis, MIT, Cambridge, MA (1987). [11] M.J. Crochet, F. Dupret, Y. Ryckmans, F.T. Geyling and E.M. Monberg, J. Crystal Growth 97 (1989) 173.