Global Analysis Of Heat Transfer Research Articles

Numerical study of heat transport and fluid flow during the silicon crystal growth process by the Czochralski method

A global analysis of heat transfer and fluid flow in a real Czochralski single silicon crystal furnace is developed using the FLUENT package. Good agreement was obtained for comparisons of the power and crystal growth speed between the simulation and experimental data, and the effect of the length of the crystal on heat transfer and fluid flow was analyzed. The results showed that Tmax increases and its location moves downward as the crystal length increases. The flow pattern in the melt does not change until the crystal grows to 900 mm. As the crystal length increases, the flow pattern in the first gas area only changes when the crystal length is less than 700 mm, but the flow pattern in the second area changes throughout the growth process.

Global analysis of heat transfer in CZ crystal growth of oxide taking into account three-dimensional unsteady melt convection: Investigation of the coupling method between 2D and 3D models

The global model of heat transfer analysis for the Czochralski crystal growth of oxides developed in our previous studies, in which a three-dimensional unsteady melt flow was taken into account, was modified. Here, the Reynolds’ averaged form of the energy equation was used for the melt in the 2D global model in order to consider the effect of unsteady three-dimensional melt convection, i.e., the heat fluxes resulting from the instantaneous three-dimensional velocity fluctuations in the melt, on the thermal field in the 2D global model. As a result, it was revealed that the effect of the time-dependent melt convection can be accounted for well by the Reynolds’ average form of the energy equation. Also, the effect of crystal rotation on melt convection and the melt/crystal interface shape was investigated numerically.

Global analysis of heat transfer in CZ crystal growth of oxide taking into account three-dimensional unsteady melt convection: Effect of meniscus shape

A global model of heat transfer analysis for the Czochralski crystal growth of oxides, in which a three-dimensional unsteady melt flow was taken into account, was developed in our recent study. In the model, however, the focal point was the methodology of formulating the model by coupling a conventional global model of heat transfer, which is based on a pseudosteady axisymmetric assumption, with a model of a three-dimensional, unsteady melt flow via two interface models. Therefore, for simplicity, the shape of the melt free surface was assumed to be flat. In this study, the global model was further developed by considering the meniscus of the melt free surface. It was found that the meniscus of the melt free surface caused the melt flow to be more unstable and shifted the critical Reynolds number at which the melt/crystal interface inversion occurs toward a much lower value.

Global analysis of heat transfer considering three-dimensional unsteady melt flow in CZ crystal growth of oxide

The conventional global model of heat transfer for the Czochralski (CZ) crystal growth of oxides is based on a pseudo-steady axisymmetric assumption. However, because oxide melt flow is commonly three-dimensional and unsteady, an approach to formulate a global model in which a three-dimensional unsteady melt flow is taken into account was proposed in this study. This approach couples a conventional global model of heat transfer and a model of a three-dimensional, unsteady melt flow using two interface models. The newly developed global model was validated and used to investigate the effect of a three-dimensional, unsteady melt flow on oxide crystal growth. The results indicate that the effect of a three-dimensional, unsteady melt flow is too large to be neglected when the crystal rotational Reynolds number is relatively large. It was found that a three-dimensional, unsteady melt flow shifts the critical Reynolds number at which interface inversion occurs at a much lower value than that obtained using a conventional model based on a pseudo-steady axisymmetric assumption.

A global analysis of heat transfer in the CZ crystal growth of oxide: Recent developments in the model

In this paper, first, we reviewed our previous studies of the global analysis of heat transfer in an inductively heated Czochralski (CZ) furnace to grow oxide single crystals, in which it was assumed that the system is in a steady state and is axially symmetric. Particularly, it was emphasized that internal radiative heat transfer within oxide single crystals is important because oxide crystals are often semitransparent to infrared radiation. The results of the global analysis revealed that the melt/crystal interface shape is strongly affected by the absorption coefficient of such crystals as well as melt convection. Next, a new simple global model, in which unsteady, three-dimensional melt convection is taken into account in addition to radiative heat transfer in such crystals, is introduced. Here, it was shown numerically that when the crystal rotational Reynolds number is relatively large, melt convection becomes unsteady and three dimensional.

01aA04 Three-dimensional global analysis of heat transfer in a unidirectional solidification process for silicon solar cells(NCCG-36)

01aA04 Three-dimensional global analysis of heat transfer in a unidirectional solidification process for silicon solar cells(NCCG-36)

01aC04 Global analysis of heat transfer with three-dimensional melt flow in CZ oxide crystal growth(NCCG-36)

01aC04 Global analysis of heat transfer with three-dimensional melt flow in CZ oxide crystal growth(NCCG-36)

Numerical simulation of the Czochralski growth process of oxide crystals with a relatively thin optical thickness

For the single crystal growth of an oxide, the global analysis of heat transfer in the inductively heated Czochralski (CZ) furnace was carried out to investigate the effect of optical properties of crystal on the CZ crystal growth process. Here, the finite volume method (FVM) was used as the radiative transfer model to solve the radiative transfer equation, and consequently the crystal with a relatively thin optical thickness (∼0.01) could be accounted for. As a result, it was found that the melt/crystal interface becomes more convex toward the melt for a small crystal rotational Reynolds number as the optical thickness of the crystal, κ s, decreases, although its dependence is slight for κ s < 0.1. In addition, the critical Reynolds number, at which the interface inversion occurs, decreases with the optical thickness of the crystal.

Variations of solid–liquid interface in the BGO low thermal gradients Cz growth for diffuse and specular crystal side surface

A global analysis of heat transfer in growing BGO (Bi 4Ge 3O 12) crystals was carried out to explain the significant variation of the solid–liquid interface shape observed in practice during crystal growth. Special attention was given to the accurate treatment of radiative heat transfer in crystal and the gap between crystal and crucible wall. Crystal side surface was assumed to be transparent and diffuse or specular (Fresnel) reflective while melt was opaque. Spectral absorptivity of crystal was approximated by a three-band model. It is shown that rotationally driven vortex dominates under the solid–liquid interface during the whole growth process and no reconstruction of flow pattern, which could lead to interface inversion, takes place. As a result, in the case of diffuse crystal surfaces, the calculated deflection of the crystallization front does not exceed several millimeters and that completely disagrees with the observations. In contrast, for the specular crystal surface the solid–liquid interface turns out to be deeply convex toward the melt at the initial stage of the growth, and then, as the crystal is pulled, its convexity strongly diminishes. Such behavior is in good agreement with the experiment and mainly determined by specular reflection at the conical part of the crystal (its shoulder) while the effect of specular reflectivity of the cylindrical part and melt-free surface is significantly less. It is also shown that radiation transfer in the wavelength band of crystal opacity in conjunction with the specular reflection of radiation in the band of crystal transparency result in the appearance of “convex–concave” isotherms in the upper part of the crystal.

Radial distribution of temperature gradients in growing CZ-Si crystals and its application to the prediction of microdefect distribution

Experiments were performed to estimate the adequacy of simulation results obtained using global analysis of heat transfer performed using melt convection models. First, we reproduced the shapes of the crystal–melt interface for several crystal lengths using two turbulence models with standard value of the Karman constant. Moreover, axial temperature distributions were measured by imbedding thermocouples into the crystal as it grew from the silicon melt. During the experiment, the crystal and the crucible were rotated to match the simulation conditions. Temperatures were measured at three points located at different distances from the crystal axis. Comparing these results with the simulation results, we found that the distribution predicted by the k− l turbulence model was more realistic and agreed well with the results of the experiments. Finally, computer simulation, based on the parallel model of the void and oxide precipitate formation in silicon crystals is reported. The unusual microdefect pattern observed in the silicon crystal with a diameter of 150 mm, where oxide particles occured in the interstitial-rich region was reproduced by the simulation and explained.

Global analysis of heat transfer in growing BGO crystals (Bi 4Ge 3O 12) by low-gradient Czochralski method

A numerical analysis of heat transfer in a set-up for growing BGO crystals is undertaken. Consideration is given to the correct treatment of radiative heat transport inside and outside the crystal. Effect of internal radiation is demonstrated. Conjugate heat transfer in crystal, melt and crucible was studied for different crystal lengths. Violation of the fully faceted growth at the initial stage of pulling is discussed.

Effect of internal radiative heat transfer on interface inversion in Czochralski crystal growth of oxides

A global analysis of heat transfer was carried out to investigate the effect of internal radiative heat transfer in the crystal and/or melt on the interface inversion in Czochralski growth of an oxide crystal. As a result, it was found that the critical Reynolds number at which the interface inversion occurs decreases with the optical thickness of the crystal when the melt is opaque. However, when the melt is semitransparent and its optical thickness is the same as the crystal's, the critical Reynolds number has a maximum value for a certain optical thickness of the crystal and melt, depending on the melt flow structure composed of the free convection and the forced convection caused by the crystal rotation.

24aA1 Investigation on various factors In the CZ crystal growth process of oxide by a global analysis of heat transfer.

24aA1 Investigation on various factors In the CZ crystal growth process of oxide by a global analysis of heat transfer.

Global analysis of heat transfer in Si CZ furnace with specular and diffuse surfaces

For the single-crystal growth of silicon, a global analysis of heat transfer in a CZ furnace was carried out using the finite element method, where the radiative heat transfer between the surfaces that possess both specular and/or diffuse reflectance components was taken into account, and then the effect of the specular reflection of the crystal and/or melt on the CZ crystal growth was numerically investigated. This was compared with the results based on the assumption of diffuse reflection which has been adopted in previous studies. As a result, it was found that the pulling rate of the crystal with a purely specular surface becomes faster than that with a diffuse surface to maintain the crystal diameter constant for a given heater power. Also, the effect of the radiative characteristic of the melt surface on the crystal growth strongly depends on the surface shape. The effect of the specular reflection should be considered in the precise numerical simulation of heat transfer in the CZ furnace, especially for a relatively large diameter, if the crystal and/or melt surfaces are specular.

Numerical and experimental studies on crack formation in LiNbO3 single crystal

A global analysis of heat transfer was carried out in an inductively heated Czochralski (CZ) furnace which was actually used to grow LiNbO3 single crystals, and then the temperature profiles obtained were used to calculate the three-dimensional thermal stress field in the crystal. By comparing the numerical results with the experimental ones, it was found that controlling the thermal environment in the CZ furnace so that the thermal stresses at the crystal surface might not exceed a certain value is important to realize the cracking free growth operation. In this study, this was accomplished through some modifications in the furnace design such as insertion of an after-heater into the furnace. These findings were verified by additional numerical simulations and crystal growth experiments for some growth conditions.

Effect of internal radiation within crystal and melt on Czochralski crystal growth of oxide

For the crystal growth of an oxide, a global analysis of heat transfer in the inductively heated Czochralski furnace was carried out, taking the radiative heat transfer within the crystal and the melt into account. As a result, the flow and temperature fields are strongly affected by the variation in optical properties of the crystal and the melt, and the melt-crystal interface shape becomes more convex to the melt as the optical absorption coefficients of both the crystal and the melt decrease.

Global Analysis of Heat Transfer in CZ Crystal Growth of Oxide.

For the crystal growth of LiNbO3, a global analysis of heat transfer in an inductively heated CZ furnace was carried out by use of the finite element method. The effects of the temperature dependency of physical properties such as viscosity and surface tension of the melt, as well as the effect of processing parameters such as crystal rotation, on the flow and temperature fields in the furnace and on the melt/crystal interface shape were investigated. With the mathematical model developed in the present work, the interface inversion caused by crystal rotation was demonstrated theoretically. It is found that the effect of the temperature dependency of viscosity on CZ crystal growth is not significant under the conditions used here, and that the thermocapillary force strongly affects crystal growth. The critical crystal rotation rate at which the interface inversion occurs becomes much larger, and the RF coil current required to keep the crystal diameter constant becomes smaller, in comparison with those without thermocapillary flow in the melt.