A quantitative method for investigating 3D oxygen transport pathways in Czochralski growth of a silicon crystal

Abstract

Elucidating complex oxygen transport is key to precise oxygen control in Czochralski (Cz) growth of a silicon crystal. We propose a quantitative method for investigating 3D oxygen transport pathways. This method can compute the origin and destination as well as the exact shape of the oxygen transport pathways in the melt. It is realized by analogizing oxygen transport to a kind of convection transport and calculating the oxygen transport ‘velocity field’. Similar to the streamlines, the pathways are obtained by numerically integrating the ‘velocity field’. Applied to the continuous-feeding Czochralski method (CCz), the results demonstrate that the shape of the pathways is approximately spiral. The rotation direction and angle of the spiral pathway are governed by the ratio of circumferential convection flux to that of axial turbulent-diffusion, while its radius evolves with the radial to axial turbulent-diffusion flux ratio. The oxygen transport pathways to the crystallization interface originate from within a circle (named ‘circle O’) at the crucible bottom, whose radius is smaller than that of the crystal and governed by the axial variation rate of the radius along the transport pathways to the three-phase contact line. This finding enables precise control of the source of oxygen transported into the silicon. Through the quantitative analysis, we also find that the inner-crucible increases the oxygen content not because of its proximity to the crystallization interface facilitating its dissolved oxygen transport into the silicon as generally accepted, but because it expands the radius of ‘circle O’ and enhances axial turbulent-diffusion. Moreover, this method is potentially applicable to quantitative studies of impurity transport for other crystal growth processes.