Fick's Laws: The Mathematics of Diffusion

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Fick's Laws: the Mathematics of Diffusion

Diffusion, the movement of a chemical species from an area of high concentration to an area of lower concentration, is one of the two major processes by which chemical species or dopants are introduced into a semiconductor (the other one being ion implantation) . The controlled diffusion of dopants into silicon to alter the type and level of conductivity of semiconductor materials is the foundation of forming a p-n junction and formation of devices during wafer fabrication. The mathematics that govern the mass transport phenomena of diffusion are based on Fick's laws.

Fick's First Law

Whenever an impurity concentration gradient, ∂C/∂x, exists in a finite volume of a matrix substance (the silicon substrate in this context), the impurity material will have the natural tendency to move in order to distribute itself more evenly within the matrix and decrease the gradient.

Given enough time, this flow of impurities will eventually result in homogeneity within the matrix, causing the net flow of impurities to stop. The mathematics of this transport mechanism was formalized in 1855 by Fick, who postulated that the flux of material across a given plane is proportional to the concentration gradient across the plane.

Thus, Fick's First Law states:

J = -D ( ∂C(x,t)/∂x )

where J is the flux, D is the diffusion constant for the material that is diffusing in the specific solvent, and ∂C(x,t)/∂x is the concentration gradient. The diffusion constant of a material is also referred to as 'diffusion coefficient' or simply 'diffusivity.' It is expressed in units of length²/time, such as µm²/hour. The negative sign of the right side of the equation indicates that the impurities are flowing in the direction of lower concentration.

Fick's Second Law

Fick's First Law does not consider the fact that the gradient and local concentration of the impurities in a material decreases with an increase in time, an aspect that's important to diffusion processes.

The flux J1 of impurities entering a section of a bar with a concentration gradient is different from the flux J2 of impurities leaving the same section. From the law of conservation of matter, the difference between J1 and J2 must result in a change in the concentration of impurities within the section (assuming that no impurities are formed or consumed in the section).

This is Fick's Second Law, which states that the change in impurity concentration over time is equal to the change in local diffusion flux, or

∂C(x,t)/∂t = - ∂J/∂x

or, from Fick's First Law,

∂C(x,t)/∂t = ∂(D∂C(x,t)/∂x)/∂x.

If the diffusion coefficient is independent of position, such as when the impurity concentration is low, then Fick's Second Law may be further simplified into the following equation:

∂C(x,t)/∂t = D ∂²C(x,t)/∂x².

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