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Mathematical Model for Diffusion
The basic one-dimensional diffusion process follows Fick's first law of diffusion:
where J is the particle flux of the donor or acceptor impurity species, N is the concentration of the impurity, and D is the diffusion coefficient.
Fick's second law of diffusion may be derived using the continuity equation for the particle flux:
This equation states that the rate of increase of concentration with time is equal to the negative of the divergence of the particle flux. For the one-dimensional case, the divergence is equal to the gradient. Combining the two above equations yields Fick's second law of diffusion:
in which the diffusion coefficient D has been assumed to be independent of position. This assumption is violated at high impurity concentrations.
The partial differential equation in the last equation can be solved by variable separation or Laplace transform techniques. Two specific types of boundary conditions are important in modeling impurity diffusion in silicon. The first is the constant-source diffusion, in which the surface concentration is held constant throughout the diffusion. The second is called a limited-source diffusion, in which a fixed quantity of the impurity species is deposited in a thin layer on the surface of the silicon.