Radiative heat exchange in Czochralski crystal growth

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.