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In this study, the effects of different rotational combinations between the crystal and the crucible on the flow, heat and mass transfer in the melt are discussed based on numerical simulations. Both crystal-crucible counter- and iso-rotations are investigated. The simulation results for the growth of crystal 200 mm in length show that the oxygen concentration along the crystal-melt interface is determined by competition between the mechanisms of convection and diffusion. When the differences in rotation between the crystal and crucible are low during iso-rotation, the melt velocity is significantly weakened and the effect of the diffusion process on oxygen transport in the melt becomes stronger than the effect of convection. The melt flow is more sensitive to changes in the crystal rotation rate. The distribution of oxygen atoms and the temperature of the melt are more significantly affected by the flow pattern in cases of counter-rotation than iso-rotation. Moreover, it is found that a flow transition occurs in the silicon melt when the crystal and crucible have the same iso-rotation rate (ReS/ReC = 0.5842). A lower concentration and more uniform radial distribution of oxygen can be obtained by using the iso-rotation conditions. Rotating the crystal and crucible in the same direction also produces a flatter defect transition with lowering and flattening of the growth parameters Vcr/Gc along the crystal-melt interface.

In this study, the power ratio between the top and side heaters and the moving velocity of the side insulation are designed to control the shape of the crystal-melt interface during the growth process of a 1600 kg multi-crystalline silicon ingot. The power ratio and insulation gap are adjusted to ensure solidification of the melt. To ensure that the crystal-melt interface is slightly convex in relation to the melt during the entire solidification process, the power ratio should be augmented gradually in the initial stages while being held to a constant value in the middle stages. Initially the gap between the side and the bottom insulation is kept small to reduce thermal stress inside the seed crystals. However, the growth rate will be slow in the early stages of the solidification process. Therefore, the movement of the side insulation is fast in the initial stages but slower in the middle stages. In the later stages, the side insulation gap is fixed. With these modifications, the convexity of the crystal-melt interface in relation to the melt can be maintained during the growth process with an approximately 41% reduction in the thermal stress inside the growing ingot and an 80% reduction in dislocation density along the center line of the ingot compared with the original case.

In this study, a global transient numerical simulation of silicon growth from the beginning of the solidification process until the end of the cooling process is carried out modeling the growth of an 800 kg ingot in an industrial seeded directional solidification furnace. The standard furnace is modified by the addition of insulating blocks in the hot zone. The simulation results show that there is a significant decrease in the thermal stress and dislocation density in the modified model as compared to the standard one (a maximal decrease of 23% and 75% along the center line of ingot for thermal stress and dislocation density, respectively). This modification reduces the heating power consumption for solidification of the silicon melt by about 17% and shortens the growth time by about 2.5 h. Moreover, it is found that adjusting the operating conditions of modified model to obtain the lower growth rate during the early stages of the solidification process can lower dislocation density and total heater power.

A three-dimensional simulation model is used to study the oxygen concentration distribution in silicon crystal during the Czochralski growth process under a transverse uniform magnetic field. The flow, temperature, and oxygen concentration distributions inside the furnace are calculated for different crystal lengths. There is significant variation in the flow structure in the melt with the growth length. The results show that in the initial stages, there is a decrease in the oxygen concentration at the crystal-melt interface as the length of the growing crystal increases. As the crystal lengthens further, a minimum value is reached after which the oxygen concentration increases continuously. This trend is consistent with that shown in the experimental results. The variation of the oxygen concentration with the growth length is strongly related to the depth of the melt in the crucible and the flow structure inside the melt. Better uniformity of the axial oxygen concentration can be achieved by proper adjustment of the crucible rotation rate during the growth process.

In this study, the installation of insulation blocks in the hot zone is utilized to assist in the growth of multi-crystalline silicon ingots with 800 kg of silicon charge using the seeded directional solidification method. A transient global numerical simulation is carried out to investigate the heat and mass transport during growth process. At a higher solidification fraction, lower concavity of the crystal–melt interface near the crucible wall can be obtained as compared to the standard model. The lowest concavity and highest energy saving is achieved when insulation blocks are added to the side of a directional solidification block and to the low part of the side insulation. The simulation results for this design also show a reduction of the melt velocity. The average oxygen concentration is slightly higher along the crystal–melt interface, compared to the standard one.

The effects of using a balanced/unbalanced cusp magnetic field (CMF) along with crystal-crucible counter-/iso-rotation on the heat, flow, and oxygen distributions during Czochralski (Cz) growth of an 8-inch silicon crystal are numerically investigated. One counter-rotation example is compared to iso-rotation cases. In both rotation modes, the vertical flow in the central melt region is strengthened while the buoyancy-driven melt flow is weakened under a cusp field. The CMF has a stronger effect on the oxygen content when there is a large difference in rotation rate between the crystal and the crucible (nS − nC). This is because a higher value of (nS − nC) induces stronger melt convection and hence, the Lorentz force has a more significant effect on a fast melt flow than a slow one. Although diffusion significantly influences oxygen transport in the melt at low crystal iso-rotation rates, different flow patterns and local melt velocities induced by different crystal iso-rotation speeds will modify the oxygen distribution. There is a slight change in the oxygen content at a low ingot iso-rotation rate with a CMF. In cases of iso-rotation, the radial oxygen content is enhanced more by an unbalanced CMF than a balanced one. Greater uniformity of the radial oxygen concentration is achieved applying iso-rotation with nS > nC under a CMF.

In this study, numerical computations are performed to predict the thermal and stress history of sapphire during the cooling process in a Kyropoulos furnace. The thermal distribution during the cooling process is controlled by the power reduction process. During the cooling process, the higher stress is located in the region near the sapphire seed crystal, the crystal center and at the crystal surfaces with large curvature. The results show that as time increases the resultant temperature gradient and the von Mises stress inside the crystal declines. The direction of heat flow at the bottom of the crucible wall changes when the heat supply from the bottom heater in the crucible is less than the heat removal from the crucible support rods. This causes a resultant temperature gradient and sudden increase in the von Mises stress inside the crystal. The highest von Mises stress in the crystal during the whole cooling process appears after this change. The fast cooling process causes a higher value of maximum von Mises stress, but releases the thermal stress more quickly.

The Kyropoulos (KY) method is commonly used to grow large sized sapphire single crystals. The shape of the sapphire crystal thus grown is determined by the heater arrangement and the power reduction history in the Kyropoulos furnace. In order to grow high-quality sapphire single crystal, the heater arrangement should allow different power inputs in different sections in order to control the thermal field in the melt during the growth process. In this study, a numerical computation is performed to investigate the effects of the heater arrangement on the thermal and flow transport, the shape of the crystal–melt interface, and the power requirements during the Kyropoulos sapphire crystal growth process in a resistance heated furnace. Four different power ratio arrangements in a three-zone heater are considered. The results show that for the power arrangements considered herein, the temperature gradients along the crystallization front do not exceed 0.05 K/mm, and that, after the growth of the crown, the crystal maintains an almost constant diameter. The remelting phenomenon may occur during growth when the input power of the upper side of the heater is higher than that of the lower side of the heater.

Numerical computation has been performed to investigate temperature and velocity distributions for different stages of the Kyropoulos sapphire single crystal-growth process. The finite-element method is employed to solve the governing equations with proper boundary conditions. In the power history considered here, a vortex appears in the melt during growth, and its strength decreases as the input power is reduced. Isotherms in the melt are distorted by flow motion. The crystal–melt interface is always convex towards the melt and in early stages the convexity increases as the input power decreases. When the crystal–melt interface is close to the bottom of the crucible, this interface is flat near the apex because of reduction in growth rate near the upper region caused by input heat from the bottom of the crucible. Therefore, convexity of the crystal–melt interface decreases the input power decreases. The crystal shape predicted by the present simulation is similar to that of crystals grown in the industry.

The effect of the RF coil position during the stages of sapphire crystal growth process in an inductively heated Czochralski crystal growth furnace on the thermal and flow transport, the shape of the crystal–melt interface shape, and the power requirements is investigated numerically. The results show that although the maximum values of temperature and velocity decrease, the convexity of the crystal–melt interface increases as the crystal length grows. It is found that the least input power is required if the central position of the RF coil is maintained below the central position of the melt during the crystal growth process. Under such crystal growth conditions, the temperature gradients along the crystalline front are small.

Erratum to ‘‘Effect of power arrangement on the crystal shape during the Kyropoulos sapphire crystal growth process’’ [J Cryst Growth, In press, corrected proof, Available online 24 January 2012] Chun-Hung Chen , Jyh-Chen Chen , Chung-Wei Lu , Che-Ming Liu c a Department of Mechanical Engineering, National Central University, No. 300, Jhongda Road, Jhongli City, Taoyuan County 32001, Taiwan, ROC b Department of Information Management, Jen-Teh Junior College, HouLung, Miao-Li 35664, Taiwan, ROC c Innovation Technology Research Center, Sino-American Silicon Products Inc., No. 8, Industrial East Road 2, Science-Based Industrial Park, Hsinchu, Taiwan, ROC