Fig 2 Constant-source diffusion results in a complementary
error function impurity distribution. The surface concentration N_{0 }remains
constant and the diffusion moves deeper into the silicon wafer as the Dt
product increases. Dt can change as a result of increasing diffusion time,
increasing diffusion temperature, or a combination of both. [1]

A constant-source diffusion or constant surface
concentration or solid solubility limited predeposition / diffusion results in
a **complementary error function**
impurity distribution. The surface concentration N_{0} remains constant
and the diffusion moves deeper into the silicon wafer as Dt product increases.
Dt can change as a result of increasing diffusion time, increasing diffusion
temperature, or a combination of both.

** **

**Initial Condition:**

** **

At time t=0, impurity concentration at depth x and time t,
that is, N(x,t) is given by,

N (x, 0) = 0

** **

**Boundary Conditions:**

At surface we have N(0,t) = N_{0} and N(¥,t) = 0 for x=¥

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

_{}

** **

Where N_{0} is the surface concentration (atoms/cm^{2})

*D* is constant diffusivity in cm^{2}/s

*x* is the distance in *cm*

* t *is the diffusion time in seconds

The table below gives some values of z= x/2(Dt)^{1/2}
and erfc(z)

z |
erfc(z) |

0 |
1.0 |

0.5 |
0.5 |

1.0 |
1.7 |

1.5 |
0.35 |

2.0 |
0.005 |

2.5 |
0.0004 |

3.0 |
0.00002 |

¥ |
0 |