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
length^{2}/time, such as µm^{2}/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 **
∂^{2}C(x,t)/∂x^{2}.

**
***See Also:
* Diffusion;
Ion
Implant;
IC
Manufacturing; Wafer Fab Equipment

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