Numerical Simulation of Thermo-elastic Stresses in Germanium Crystal Grown by Czochralski Technique during Different Lengths of Crystal

In the process of crystal growth by Czochralski technique, lower part and core of the crystal are warmer than other parts of crystal and its environment, which leads to expansion in different parts of the crystal. The result of this thermal gradient is strain, which eventually causes thermo-elastic stress in the crystal. Increasing this stress leads to transition of the material from elastic limit and entering plastic area. To show thermo-elastic stress in crystals, a criterion called Von Misses stress is used. Using the solid mechanics approach, the mechanical response of crystal to the stresses can be determined through appropriate structural equations. In this paper, using appropriate structural equations, a set of numerical simulations of temperature field, thermal stress and dislocation density for a Czochralski setup used to grow Ge single crystal have been done for different heights of crystal. In order to investigate dislocation density, using a simple first-order approximation, in which the dislocation density is proportional to radial gradient of temperature is used. A two-dimensional steady state finite element method has been applied for all calculations. The numerical results reveal that the thermal field and thermal stress are mainly dependent on the crystal height, heat radiation and gas flow in the growth system. As the height of the crystal increases and the shape of the crystal-melt interface changes, we see an increase in thermo-elastic stress and dislocation density.