Thermal Field Simulation of LEC Process by Finite Volume Method

Copyright ID IFAC Dynamics and Control of Process Systems, Corfu, Greece, 1998

THERMAL FIELD SIMULATION OF LEC PROCESS BY FINITE VOLUME METHOD Min-ho Suh and Tai-yong Lee

Department of Chemical Engineering Korea Advanced Institute of Science and Technology 373-1. Kusong-dong. Yusong-gu, Taejon. KOREA e-mail: [email protected]@che.kaist.ac.kr

Abstract: Thermal field is obtained for the Liquid Encapsulated Czochralski process by Quasi-Steady State Model. Our model includes the effects of convective heat transfer in the melt and radiational heat transfer of the equipment surfaces. To calculate the radiational heat flux , the view factor calculation is performed. Encapsulant layer is assumed to be totally opaque or transparent to the radiation. The thermal field formation and the temperature of the heater depend on the thermal transparency of the encapsulant layer. The effects of melt flow and thermal transparency of the encapsulant layer are analysed. The proposed model can be extended to the semi-transparent model and used as thermal field prediction model for optimization of LEC process. Copyright © 1998lFAC Keywords: Crystal Growth. LEC process. QSSM . FVM. Modeling & Simulation

pared with CZ process to prevent the volatilization of As component in the melt. The thermal transparency of the encapsulant layer to the radiational heat make the heat transfer mechanism more complex . The characteristics of grown single crystal are greatly affected by the thermal transparency of encapsulant layer . The complexity of heat transfer in the LEC process must be reflected by the model. We concentrated on the effects of the thermal transparency of the encapsulant layer to the thermal field formation. Our model reflects the effects of the melt flow in the melt, thermal transparency of encapsulant layer. and the radiational heat flow in system. The quality of single crystal is usually evaluated by dislocation density. It is hard to measure dislocation density on line. LEC process needs the gain scheduling of heater power controller to meet the dislocation density constraint of crystal quality. Because the dislocation is formed mainly by the thermoelastic stresses. it is very important to predict the temperature field in LEC process. We concentrated

1. INTRODUCTION Gallium Arsenide( GaAs) is basic material for manufacturing the digital Large Scale Integration(LSI) circuit. and the application range is extended to the light-electronic devices and microwave devices . Advance in GaAs application fields depends on producing the low dislocation density wafer of homogeneous and stable charateristics. It is well known fact that the advanced technique in Czochralski( CZ) process is not directly applicable to Liquid Encapsulated Czochralski(LEC) process (Muller et al.. 1992). GaAs crystal has lower thermal conductivity than Silicon(Si) crystal. The thermal gradient in the crystal is large. and thermoelastic stresses are also large . (Jordan et al.. 1981) has first identified that the high thermoelastic stresses are the main problem in growing the single crystal in CZ related process. One of the main objectives in optimization of single crystal growth process is reducing the thermoelastic stresses. Also. the LEC process has the additional encapsulant layer(B 2 0 3 ) com-

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Sta inless steel chamber

Ar gas

I I ~ jV+

~ D.

D,

D.

D,

D,

(b) transparent

ta) opaque

Fig.

2. Radiation heat exchange (a)opaque; (b )transparent

domain:

to be growing with constant radius and the time scale of melt height(H) change is much larger than that of thermal heat transfer. QSSA is justified by equation (1).

Z t~ characteristic time for heat transfer characteristic time for melt volume change

r Fig. 1. Schematic diagram of the LEC process

L 2 /o. = L/V+

on modeling the temperature field in order to use it as optimization tool of process condition. Single crystal growth system is inherently batch process. We adopt the Quasi-Steady State Approximation(QSSA) in modeling the thermal field of LEC process than the fully transient approach , because QSSA can reduce the computational load efficiently, but give us the useful information for the gain scheduling of heater power control to reduce the dislocation induced by the thermoelastic stresses . If a constant-diameter crystal is grown , QSSA can be successfully exploited to describe the LEC system (Derby et al., 1988). Justification of QSSA is explained in next section. This modeling is basic work for linking the thermal field and thermoelastic stresses field . With thermal field model and thermoelastic field model, we can find the optimal heater temperature profile to reduce the dislocation density in LEC process.

= 0.05

«

1.0

(1)

Where L = 0.1016m is the charateristic length of the system , 0. = 5.84 x 10- 6 m 2 / s is the thermal diffusivity of the melt , and V+ = 0.01 m/ hr is the pulling rate . Then it is valid to reconstruct the time history of a growth run from a sequence of steady-state calculations with decreased melt depths . So we obtained three steady-state solutions increasing the crystal height , and assumed that each solution represents the initial , middle , last stage of crystal growing process .

2.2 Governing Equations

Conduction governs the heat transfer in crystal(Ds). encapsulant(D E), crucible(Dc) , suspector(D p) part. But in melt phase(DM), the convective heat transfer by the melt flow is very important. So the dynamics of melt flow in the crucible are modeled by Navier-Stokes equation . Natural convection is represented by Boussinesq assumption.

2. MODELING 2.1 Quasi-Steady State Approximation

i

We used the Quasi-Steady State Model(QSS1f) . The crystal is pulled continuously from the melt at a rate F+ (Fig. 1). After the initial seeding and shouldering process, the crystal is assumed

PeM (u · \7 0

M)

u ' \'u

(2)

2

(3)

\7 0 M

= V' . a +RaPr0 Me

\7·u=O

524

=

= S,E . C , P z

(4) (5)

1.2. . 5

r------------------" Ar gas

T 1226.2 1261.8 1297.4 13330 1368.6 1404.2 1439.8 1475.4 1511.0 1537.7 1564 .4

, 1&+5

1.0&+5

~

I

90&+4

n

~

c

n

~

ro

.!:l

'"t'

:;

"

" ~

~

g, melt

8 .09+.

(a) opaque 7 .0&+4

6 .0&+4 .JL--~-____,--_,____-~-~------l 60&+4 7.0&+4 8 .08... 9.Cle+4 1.08-+5 1.18+5 1.29-+5

T 1386.4 1404.2 1422.0 1439.8 1457.6 1475.4 1493.2 1511.0 1524.35 1537.7

Heal_Generation(WaU)

Fig . 3. Energy balance of the LEC process Where 0, is dimension less temperature of i domain , ii is dimensionless velocity vector in melt phase, eo is unit vector of z-direction , PeM is Peclet number of melt phase, Ra is Rayleigh number, and Pr is Prandtl number.

Ar gas

n ~

c

n <:]"

;;-

~v.

>,

~

'-

~

"

" ~

~

melt

.'

(b) transparent Fig. 4. Thermal field of the encapsulant region: (a)opaque; and (b )transparent

2.3 Boundary Conditions

influence of scalar variables such as pressure and temperature properly.

To implement the thermal boundary conditions with radiational heat flux at the equipment surfaces, we calculated for fifteen surfaces of finite area. First. we calculated the view factors of radiational heat transfer enclosure with fifteen finite areas, and used the Gebhart energy absorption method (Gebhart , 1989). The radiational heat transfer enclosure is defined differently by the thermal transparency of the encapsulant layer. We simulate two opposite cases totally transparent and totally opaque to examine the effect of the encapsulant layer. The Fig. 2 shows enclosure definitions of two opposite cases .

3.2 QUICK

When we solve the convection-conduction problem, the coefficients of the FVM discretied equation always have the possibility of negative value. All the coefficients must have positive value in every iteration . Quadratic Upstream Interpolation for Convective Kinetics(QUICK) (Hayase et al., 1992) technique is adopted to guarantee the convergence and physical reliability of solution . The accuracy of QUICK scheme is second-order in terms of Taylor series truncation error.

3. NUMERICAL METHOD

The Finite Volume Method(FVM) is used to discretize the governing equations. We used SIMPLER (Pantankar, 1980) algorithm to get the iterative solution of discretised equations with QUICK scheme. Numerical solution is checked by the whole energy balance of the system.

3.3 SIMPLER algorithm

SIMPLER is for solving momentum , or energymomentum coupled partial differential equations by iterative method. In case that boundary conditions of momentum calculation domain are given by velocities of fluid , we cannot know the boundary conditions of pressure. In this case it is more effective to solve continuity and momentum equation sequentially and get the converged solution by iterative method. We obtained the heater temperature satisfying the heat balance at the meltcrystal interface. At the interface, the phase transition from liquid to solid exists with latent heat generation. The heater temperature is corrected by the error of heat balance at the interface including the latent heat in solution algorithm.

3.1 FVM The most attractive feature of FVM formulation is that the resulting solution would imply that the integral conservation of quantities such as mass, momentum. and energy is exactly satified over any group of control volumes and , of course, over the whole calculation domain. We used the staggered grid for the velocity components to represent the

525

1720

1680

ir================:;----i """"9- rlled;,od lr:l(\~(lMcnl ( C".\ C 1

-0- (ll(OItlOllil :..nd

ID )

l1i1l1~r;arcn( rC .. .;,e IV )

11

~

.55

1.00

1.40

2.00

Crysla) Heighl

Fig. 5. Heater temperature variation as the crystal growth

4 .2 Temperature of the heater

Fig. 5 shows the temperature variation as the growth process going on. The temperature of the heater is high at the initial stage but low at the middle stage, and increases again . To explain this phenomena. We examine the radiational heat loss from the crystal surfaces and crucible inner wall . The temperature of the heater was qualitively proportional to the sum of the heat loss from the crystal surfaces and the crucible inner wall. We can know that heat loss from the equipment surfaces at the given radiational heat transfer configuration determines the heater temperature that maintain the given steady-state condition. And radiational heat transfer governs the whole heat transfer in the system because operation temperature of the LEC process is very high.

3.4 The whole energy balance

4.2.1. 'opaque' vs. 'transparent ' The energy balance for the whole system is checked(Fig. 3). The system boundary is the chamber wall. The heat is generated from the heater surface and melt-crystal interface by phase transition. The heat flows out through chamber wall. The relative error of the whole energy balance is 0.1 - 1.0 %. The obtained thermal field and velocity field are physically reasonable.

In case that the encapsulant is assumed to be transparent , the heat supply by radiation from the submerged part of the crucible inner wall to the submerged crystal surfaces exists , so the temperature of the crystal near the melt-crystal interface is maintained high . the heat flux through the interface is low, that mean temperature of the heater can be lower than that of opaque case. But in case of opaque, the temperature near the interface is maintained low , so the flux is high , and the temperature of the heater must be high

4. RESULTS AND DISCUSSION

4.2 .2. 'rotating ' vs. 'fixed '

First, the effects of thermal transparency of the encapsulant layer to the thermal field formation is analyzed . Not only the thermal transparency of the encapsulant but also the rotational condition affect the temperature of the heater. Velocity fields as the rotational condition of the crystal and crucible are characterized.

In case that the crystal and crucible rotates in opposite direction , the first step of heat transfer near the crucible wall is governed by only conduction . But In case that the crystal and crucible are fixed, the active convective heat transfer in the melt phase assist the heat transfer from the crucible wall to the melt-crystal interface, so the the temperature of the heater can be lower than that of rotating case.

4.1 Thermal transparency of encapsulant

4.3 Velocity field

Fig. 4 shows the thermal field of the encapsulant region as the thermal transparency of the encapsulant layer. The thick line indicates 1511K isotherm. In opaque case. the isotherm is horizontal and the heat flows from the melt-encapsulant interface to the encapsulant-Ar gas surface. In transparent case. the isotherm is inclined 45° to the vertical direction, so the heat flows from the submerged crucible inner wall to the crystal surface. The temperature of the crystal near the meltcrystal interface is maintained high in transparent case compared with in opaque case. Also the axial temperature gradient is lower than that of opaque case.

Fig. 6 shows the features of the velocity field as the rotation condition . Isotherm spacing is 10K above melting pOint(1511K) and 50K below the melting point. In the case that the crystal and the crucible rotate in opposite direction , there is almost no meridional flow near the wall and the velocity fields looks like the rigidbody rotating motion. The azimuthal motion in the melt which does not assist in heat transfer ; hence , the isotherms resemble the conduction only model driven thermal fields. In contrast to that case, when the crystal and the crucible are fixed, the buoyancy

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field and velocity field. Surface temperature of the heater in the given radiative heat transfer configuration in the process has strong interaction with the radiational heat flow from the crystal surface and crucible inner wall. Thermal transparency of the encapsulant layer affects the temperature of the crystal near the interface, and the temperature of the heater surface and thermal field in the encapsulant region depends on the thermal transparency of encapsulant.

5. CONCLUSION Energy balance for the whole system is performed to check the reliability of numerical solution. Our model satisfies the important heat balance at the melt-crystal interface because it is implented by the conversion criterior of the heater temperature. Also the fixed flat geometry of the melt-crystal interface reduce the calculation load. Proposed model describing thermal field is easily applicable to next steps such as analysis of thermal stress in the growing crystal , based on assumption of cylindrical crystal geometry, but realization of the crystal and melt interface geometry in the model is indispensable to be used as a tool of process optimization.

(a) rotating condition

6. ACKNOWLEDGEMENT This research was financially supported by Korea Science and Engineering Foundation through Automations Research Center and Project name 'Process design for semiconductor fabrication ' from KAIST.

(b) fixed condition Fig. 6. Thermal field and velocity field: (a) VOcrysta( -16 rpm, VOcrucible: 8 rpm; (b) VOcrystal: 0 rpm , 'V!Jcrucible: 0 rpm

7. REFERENCES

driven flow help the heat transfer in the melt, and the isotherm shows the flow is going up near the crucible wall and down in the center.

Gebhart, B. (1989). Heat transfer. McGraw-hill. Hayase, T .. J. Humphrey and R Greif (1992). 1. of Computational Physics 98. 108-118. Jordan , A.S .. R Caruso and A.R. VonNeida (1981). J. Applied Phisics 52, 3331-3336. Derby, J.J., RA. Brown (1981) . J. Crystal Growth 87.251-260. Muller. G., G. Hirt and D. Hofmann (1992). 7th Conf. On Semi-insulating Materials. Mexico 3, 73-84. Pantankar. S.V. (1980). Numerical heat transfer and fluid flow. McGraw-hill.

Thermal field is obtained for the Liquid Encapsulated Czochralski process that produces commercial scale 4-inch GaAs wafer by Quasi-Steady State Model. The heat transfer mechanism of the model includes the radiational heat transfer of the equipment components' surfaces and convectional heat transfer in melt flow of the crucible. Encapsulant layer is assumed to be totally opaque or transparent. Solutions of the model equations are obtained for the cases in which the crystal and crucible are stationary, as well as they rotate in opposite direction. The surface temperature of the heater satisfying the heat balance including the latent heat generation at the crystal and melt interface is obtained simultaneously with thermal

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