Fig. 3 A **gaussian
distribution** results from a limited-source diffusion or constant total
dopant diffusion. As the Dt product increases, the diffusion front moves more
deeply into the wafer and the surface concentration decreases. The area under
each of the three curves is the same. [1]

Limited-source diffusion is modeled mathematically using
an impulse function at the silicon surface as the initial boundary condition.
The magnitude of the impulse is equal to the dose *Q.*

* *

* *

* *_{}

**Initial condition:**

** **

At t=0, N (x, 0)=0

**Boundary Conditions:**

** **

_{}_{} And C (¥, t) = 0

**The solution** **of Fick's second law that satisfies the initial and boundary conditions
is given by:**

** **

_{}

** **

**By setting x = 0 we
obtain the surface concentration**

** **

_{}

** **

**The solution can be
rewritten as **

** **

_{}