Bulk growth of semiconductor crystals in a magnetic field: A study of dopant transport

When semiconductor crystals are grown from a body of liquid (melt), the turbulent melt motion produces spatial oscillations of the dopant concentration in the crystal, which are called striations or microsegregation. Dopants are elements which are added to the melt in order to change the electrical or optical properties of the crystal. The acceptable variation of the dopant concentration in the striations has decreased dramatically as the size of devices in the integrated circuits produced on single-crystal wafers has decreased. Since molten semiconductors are good electrical conductors, a weak magnetic field is sufficient to eliminate turbulence and other unsteadiness in the melt motion. Unfortunately the elimination of turbulent mixing may lead to a large-scale variation of the crystal's dopant concentration, which is called macrosegregation. An accurate predictive tool is needed in order to tailor the magnetic field and to adjust other process parameters so that the crystal has both microscopic and macroscopic uniformity in its dopant concentration. This thesis presents one such predictive tool.;Our model for the unsteady transport of a dopant during the entire period of time required to grow a crystal assumes that the externally applied magnetic field is sufficiently strong that inertial effects and convective heat transfer are negligible. We divide the semiconductor melt into (1) mass-diffusion boundary layers where convective and diffusive mass transfer are comparable, and (2) a core region where diffusion is negligible, so that the concentration of each fluid particle is constant. A Lagrangian description of motion is used to track each fluid particle during its transits across the core between diffusion layers. The dopant distribution in each layer depends on the concentrations of all fluid particles which are entering this layer. The dopant distribution is very non-uniform throughout the melt and is far from the instantaneous steady state at each stage during crystal growth. Our transient model is the first model to predict the dopant distribution in the entire crystal. The predictions of this asymptotic model are confirmed by a numerical solution to the full mass transport equation.