ON THE PREDICTION OF DISLOCATION FORMATION IN SEMICONDUCTOR CRYSTALS GROWN FROM THE MELT - ANALYSIS OF THE HAASEN MODEL FOR PLASTIC-DEFORMATION DYNAMICS

The Haasen model for plastic deformation by the formation of dislocations in diamond structure semiconductor crystals is analyzed for thermal stress fields that are indicative of liquid encapsulated Czochralski growth of InP and GaAs and for Czochralski growth of silicon. For typical crystal growth conditions, the rapid rate of dislocation generation compared to the rate of motion of the crystal through the thermal stress field leads to asymptotic expressions for the bulk dislocation density that depend on the variation of the von Mises stress sigma(z) in the z direction of crystal growth. The dislocation density is predicted to scale as sigma-0(2) if the largest stress occurs at the melt/crystal interface and as delta-sigma-2(z) if the stress increases with distance above the interface, where sigma-0 is the value of the stress at the interface and delta is a measure of the back stress created by the dislocations. These asymptotic results agree quantitatively with the numerical calculations of Volkl and Muller using the Haasen model for prediction of the dislocation density in InP growth and give simple criteria for decreasing the dislocation density in LEC growth of InP and GaAs. Predictions for silicon growth do not agree with the experimentally observed dislocation-free crystals; possible sources for this discrepancy are discussed.