Crystal shape 2D modeling for transient CZ silicon crystal growth

• A non-stationary mathematical model of Czochralski process is presented.
• PID-based process control is considered.
• Heater is modeled with a simplified integral model.
• Unstructured finite element mesh and automatic element size control are used.
• A simulation of start cone growth is given and discussed.

A non-stationary axisymmetric model of Czochralski silicon single crystal growth is presented. The model describes transient behavior of crystal–melt, melt–gas and crystal–gas interfaces in connection with PID-based control of crystal diameter by changing crystal pulling velocity and heater power. To calculate significant crystal shape changes, unstructured finite element mesh is used in crystal and melt together with automatic element size control. Heater temperature changes are modeled with a simplified integral model. A numerical simulation example of start cone growth is given.