Jeffrey J. Derby

Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth. I. Simulation

Abstract Results are presented from finite element analysis of the Czochralski (CZ) and Liquid Encapsulated Czochralski (LEC) crystal growth processes based on a thermal-capillary model which governs the heat transfer in the system simultaneously with setting the shapes of the melt/solid interface, the melt and encapsulant menisci, and the radius of a steadily growing crystal. Calculations are performed for the small-scale growth of silicon (CZ) and gallium arsenide (LED). The effects of melt volume and crucible position relative to the heater on the radius of the crystal and the shape of the melt/solid interface are predicted for the CZ system, and the importance of including an accurate representation of the melt meniscus for modelling the process is demonstrated. The additional effect of an encapsulant layer on heat transfer is treated for the LEC method for the cases of totally transparent and opaque encapsulant. The responses of these LEC prototype systems are examined for changes in pull rate and encapsulant volume.

Heat transfer and interface inversion during the Czochralski growth of yttrium aluminum garnet and gadolinium gallium garnet

Abstract Computer-aided simulations are performed using an integrated process model for the Czochralski (CZ) growth of yttrium aluminum garnet (YAG) and gadolinium gallium garnet (GGG). Internal radiant heat trabsfer through the crystal is responsible for the deeply convex melt/crystal interface and the propensity for crystal cracking with large cone angles during CZ YAG growth. The nature of interface inversion by crystal rotation is fundamentally different for YAG growth under low and high thermal gradients. Results suggest that classical “flat-interface” growth via crystal rotation is attainable for YAG growth only under low-gradient thermal conditions, while this limitation is not as stringent for the growth of GGG. The depth of the melt is also shown to affect interface inversion for GGG. The complicated dependence of interface inversion on many system details, rather than solely crystal rotation, suggests that this effect is not adequately described by simple, universal scalings which have been previously proposed.

Heat transfer in vertical Bridgman growth of oxides: Effects of conduction, convection, and internal radiation

Abstract The vertical Bridgman growth of an oxide crystal with properties chosen to resemble those of yttrium aluminum garnet (YAG) is investigated. Internal radiation and heat conduction are accounted for in the crystalline phase, while transport through the melt (which is assumed opaque) is dominated by convection and conduction. Heat is also conducted through the ampoule walls, whose outer surface exchanges energy with the furnace via combined natural convection and enclosure radiation. A quasi-steady-state, axisymmetric Galerkin finite element method is employed for the calculation of thermal fields, melt flow patterns and melt/crystal interface shapes and positions for different parameter values. Results indicate that heat transfer through the system is strongly affected by the optical absorption coefficient of the crystal and that convective heat transfer through the melt is unimportant for this small-scale system. Coupling of internal radiation through the crystal with conduction through the ampoule walls promotes melt/crystal interface shapes which are highly deflected near the ampoule wall. This radiative interface effect is much more pronounced than that observed in the Bridgman growth of opaque crystals, where the interface deflection at the ampoule wall is attributed to the thermal conductivity mismatch between ampoule and charge. Calculations demonstrate that a flatter overall interface shape can be achieved through optimization of ampoule properties and furnace temperature profiles.

An integrated process model for the growth of oxide crystals by the Czochralski method

Abstract A comprehensive model for the Czochralski (CZ) growth of oxide crystals is developed which integrates analyses of global heat transfer, transport phenomena in the crystal and melt, and the interfaces of the growth system. Heat transfer at the global scale includes fundamental descriptions of induction heating and diffuse-gray enclosure radiation. In the crystal and the melt, steady-state axisymmetric solutions for heat transfer and fluid mechanics are computed along with a self-consistent description of the free-boundaries of the melt/crystal interface, the melt meniscus, and the crystal diameter. A Galerkin finite-element method is employed to discretize the model equations, and solutions are obtained using a quasi-Newton iterative scheme. Results are presented for the growth of a large-dimension oxide crystal with realistic thermophysical properties similar to those of gadolinium gallium garnet (GGG). Comparison between the results of this model and those of Sackinger et al. [Intern. J. Numer. Methods Fluids 9 (1988) 453] demonstrate the importance of realistic heat transfer boundary conditions. Calculations also show the effects of pedestal heat transfer on the flow in the melt. The effect of changing melt volume is examined to assess the batchwise evolution of a CZ growth run.

Modeling the vertical Bridgman growth of cadmium zinc telluride. I: Quasi-steady analysis of heat transfer and convection

A quasi-steady-state analysis of the vertical Bridgman growth of large-diameter, cadmium zinc telluride is conducted using a finite element model which accounts for the details of heat transfer, melt convection, and solid/liquid interface shape. Large radial gradients are shown to dominate the temperature field in the solid, while convection flattens the radial temperature distribution in the melt. Concave interface shapes are predicted to arise from the thermal conductivity mismatch between solid and liquid. The shape of the solid/liquid interface is sensitive to the growth rate due to the importance of latent heat release.

On the dynamics of Czochralski crystal growth

Abstract The linear and nonlinear dynamics of Czochralski (CZ) crystal growth are analyzed by a thermal-capillary model which accounts for transients caused by energy transport, by changes in the shapes of the phase boundaries, and by the batchwise decrease of the melt volume in the crucible. Solutions are computed by a finite-element/Newton method and fully implicit time integration. Solutions for a quasi-steady-state model (QSSM) are calculated for long crystals with the melt volume held. Linear stability analysis of the dynamic model shows that the QSSM solutions are stable to random small disturbances, thereby confirming the inherent stability of the CZ process. Proportional and integral feedback control strategies are tested for on-line control of the crystal radius by integration of the dynamic model augmented with a servo-control equation. Integral control leads to oscillations in the crystal radius that are not present when the controller has a proportional element.

Effect of accelerated crucible rotation on melt composition in high-pressure vertical Bridgman growth of cadmium zinc telluride

Abstract Three-dimensional axisymmetric, time-dependent simulations of the high-pressure vertical Bridgman growth of large-diameter cadmium zinc telluride are performed to study the effect of accelerated crucible rotation (ACRT) on crystal growth dynamics. The model includes details of heat transfer, melt convection, solid–liquid interface shape, and dilute zinc segregation. Application of ACRT greatly improves mixing in the melt, but causes an overall increased deflection of the solid–liquid interface. The flow exhibits a Taylor–Gortler instability at the crucible sidewall, which further enhances melt mixing. The rate of mixing depends strongly on the length of the ACRT cycle, with an optimum half-cycle length between 2 and 4 Ekman time units. Significant melting of the crystal occurs during a portion of the rotation cycle, caused by periodic reversal of the secondary flow at the solid–liquid interface, indicating the possibility of compositional striations.

The role of internal radiation and melt convection in Czochralski oxide growth: deep interfaces, interface inversion, and spiraling

Abstract Calculations are performed with an integrated process model for Czochralski oxide growth in which internal radiative heat transfer through the crystal is approximated. For the first time, melt/crystal interfaces are predicted which are deeply convex toward the melt and similar in shape to those observed experimentally. These results provide additional support for the importance of internal radiation in these systems. The inversion of these deep interfaces is also examined as a function of forced and natural convection in the melt. Finally, a new mechanism for the onset of crystal spiraling is postulated from observations of superheated crystal regions in some calculations.

Radiative heat exchange in Czochralski crystal growth

Abstract The effects of diffuse-gray radiation on the parametric sensitivity and stability of the Czochralski process for growing silicon is analyzed in a thermal-capillary model which governs heat transfer in the system, the shape of the melt/crystal and melt/gas interfaces and the shape of the growing crystal. Calculations with a quasi-steady-state model (QSSM) demonstrate differences in sensitivity of the crystal radius and the shape of the crystal/melt interfaces to melt volume and pull rate for systems with high and low axial temperature gradients, as set by the difference between the heater and ambient temperatures. The stability of these states is demonstrated by transient simulations of the response of the system to step changes in the pull rate. The crystal radius exhibits decaying oscillations which damp more slowly when the temperature gradient is low, indicating the incipient instability expected for an isothermal system. The oscillations are induced by the interactions of radiation with the shape of the crystal and are nor predicted when an idealized model is used which ignores this effect. Batchwise simulations which include the decreasing melt volume show the importance of including detailed radiation in modelling the entire growth process. Proportional and integral feedback control strategies are demonstrated for control of the crystal radius by incorporating a servo-control equation for the heater temperature into the dynamic simulation based on the thermal-capillary model.

Internal radiative transport in the vertical Bridgman growth of semitransparent crystals

Abstract A quasi-steady-state model for the vertical Bridgman growth of semitransparent (partially absorbing and emitting) crystals is developed. For the first time, internal radiative heat transfer through the crystal (considered to be a participatine medium) is calculated rigorously, taking into account the full three-dimensional shape of the crystal, including curvature of the melt/crystal interface. The crystal is assumed to be grey, and the ampoule walls, as well as the crystal/melt interface, are assumed to be grey and diffuse. Heat transfer through the melt is considered to be dominated by conduction. A Galerkin finite element method is employed to determine the position and shape of the interface and the temperature field in the crystal and melt. Results for a model oxide growth system demonstrate the sensitivity of melt/crystal interface shape, position, and interfacial gradients to changes in optical properties. The total heat flow through the melt and crystal also depends strongly on the optical parameters of the system. The phenomenon of radiative supercooling is not observed for the system considered here.