Finite Difference Analysis of Radial Phosphorus Dopant Distribution in Czochralski-Grown Silicon Single Crystals

The electrical properties of semiconductor devices are directly related to the dopant concentration in single crystals. Therefore, to make semiconductor devices with controllable electrical properties, it is important to control the dopant concentration accurately, both along the growth axis and in the radial direction. The dopant concentration along the growth direction can be described by the normal freezing equation and is well understood, but the factors controlling the dopant distribution in the radial direction are not well understood. We, therefore, made a detailed, quantitative analysis of the radial dopant distribution in Czochralski silicon crystals by solving the coupled Navier-Stokes, continuity, and energy equations for the silicon melt flow and temperature fields, and by solving the diffusion and segregation equations for the phosphorus distribution in the melt and in the crystal. Good agreement between measured and simulated results of the radial phosphorus concentration in silicon single crystals was obtained. The melt was exposed to a background gas of Ar into which was added to counteract evaporation of the phosphorus from the melt. Simulated radial phosphorus concentration distributions compared well with measured radial distributions, and the added to the background Ar gas increased the average melt dopant concentration and also improved the radial phosphorus concentration uniformity.