**High-pressure liquid-encapsulated
Czochralski growth (HPLEC) of InP**

Unlike the conventional Czochralski
process, modeling of high-pressure liquid-encapsulated Czochralski
(HPLEC) method is particularly complicated, because
account must be taken of turbulent gas and melt
convection as well as of the encapsulant preventing phosphorus evaporation
from the melt. The heat transfer interactions between the recirculating gas, the
B_{2}O_{3} layer and the crystal is very complex
and requires especially accurate simulation.

(a) |
(b) |

Fig. 1. Computational domain for
preliminary 2D steady-state simulation of global heat transfer with the computed temperature
distribution and flow pattern (a), computational domain for detailed 3D unsteady
simulations (b). |

To study the influence of the gas flow on the
thermal field in the crystal, we used a 3D unsteady model that can
directly resolve turbulent eddies and their
contribution to the heat exchange. We considered gas
convection together with heat transfer in the crystal and encapsulant flow
in an industrial HPLEC InP set-up, paying special attention to the temperature
gradients in the crystal as they determine the thermal stresses.

The computational procedure included preliminary simulation of global
heat transfer in the whole set-up with the account of all the heat exchange
modes — convective, radiative and conductive — performed using a 2D
axisymmetric steady-state model. The resulting thermal field was used
to set boundary conditions for the 3D computational domain, see Figure 1.

10 sec |
20 sec |

Fig. 2. Instantaneous flow patterns and temperature
distributions in the gas domain at two different time moments. |

The results demonstrate an essentially
non-axisymmetrical flow structure with multi-scale vortices
in different areas of the computational domain.
The solution is generally characterized by
an upward flow along the pulling rod, a downward flow
along the cool external wall and two re-circulating zones in
the area between the crystal and the crucible, with the most
intensive unsteady convection observed at the center of
the growth set-up, around the crystal and the pulling rod,
see Figure 2.

(a) |

(b)
Fig. 3. Instantaneous distributions of the absolute (a) and radial
(b) temperature gradients at the two time moments. |

Strong temperature fluctuations in the gas result in pronounced variations
of the temperature and temperature gradients at the crystal periphery.
One can see in Figure 3 that the largest absolute gradients take place
at the melt/crystal/encapsulant tri-junction point, while the area,
where a large radial gradient occurs, is located at the
crystal/encapsulant/gas tri-junction point. Besides, the fluctuations
of the radial temperature gradients in the conic part of the crystal can
give rise to multiplication of dislocations.

To estimate the contribution of gas convection to the heat loss from
the crystal surface, we compared time-averaged heat fluxes across
the crystal and the encapsulant in different regimes, Figure 4. It can be seen that the heat loss from
the crystal and the encapsulant increases dramatically with the gas pressure
due to the effect of turbulent gas convection. Similar computations
made within a 2D quasi-steady axisymmetrical approach, using
two various turbulence models, significantly underestimated the heat
loss from the crystal and the encapsulant.
Summarizing,
the nitrogen flow around the crystal greatly influences the axial temperature
gradient in the crystal and, furthermore, produces observable
unsteady fluctuations of the radial temperature
gradients in the conic part of the crystal, which can
result in significantly higher dislocation density. It is therefore important
to take accurate account of unsteady gas convection in an analysis of
defect formation and optimization of the growth process. Since
2D models usually strongly underestimate the gas cooling of the crystal,
the 3D unsteady analysis is
necessary for the adequate description of the convective gas effect.

(a) |
(b) |

Fig. 4. Average heat fluxes across
the crystal (a) and encapsulant (b) surfaces, computed with the 3D unsteady (solid lines)
and 2D steady (dashed and dash-dotted lines) models. |

1. "Prediction of the melt/crystal interface geometry in liquid
encapsulated Czochralski growth of InP bulk crystals",
E.N. Bystrova, V.V. Kalaev, O.V. Smirnova, E.V. Yakovlev, Yu.N. Makarov,
J. Crystal Growth, 250 (2003) pp. 189–194.

2. "3D unsteady analysis of gas turbulent convection
during HPLEC InP growth",
E.N. Bystrova, V.V. Kalaev,
J. Crystal Growth, 287 (2006) pp. 275–280.