Diffusion can be defined as the random walk of an ensemble of particles from regions of high concentration to regions of lower concentration. In integrated circuit fabrication, diffusion is used to introduce dopants in controlled amounts into the semiconductor substrate. In particular, diffusion is used to:
According to the First Law of Diffusion, the transfer of solute atoms per unit area in a 1-dimensional flow can be described by the following equation:
where J is the particle flux, C is the concentration of the solute, D is the diffusion coefficient, x is the distance into the substrate, and t is the diffusion time. The negative sign indicates that the diffusing mass flows in the direction of decreasing concentration.
From the Conservation of Mass, we also know that:
If we combine this relationship with the 1st Law of Diffusion, then we have derived the 2nd Law of Diffusion (otherwise known as Fick's Law), which states:
In order to solve Fick's Law, one initial condition and two boundary conditions are required. Two solutions to Fick's Law are generally encountered in IC fabrication: infinte-source and limited-source diffusion. These are each described below.
Infinte-source diffusion requires a constant surface concentration of diffusing atoms. This generally corresponds to the process step known as "pre-deposition." In this case, the initial condition and boundary conditions are:
where Cs is the surface concentration. The solution to Fick's Law under these conditions is:
where "erfc" is the complementary error function.
where S is called the "dose." The solution to Fick's Law under these conditions is:
We are often interested in solving Fick's Law to determine the position of the metallurgical junction formed between a p and n region after diffusion takes place. This junction is defined as the point where C(x) is equal to the substrate concentration.